Efficient near-field wireless energy transfer using adiabatic system variations

ABSTRACT

Disclosed is a method for transferring energy wirelessly including transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ 1B , transferring energy wirelessly from the intermediate resonator structure to a second resonator structure, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ B2 , and during the wireless energy transfers, adjusting at least one of the coupling rates κ 1B  and κ B2  to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.

CROSS REFERENCE TO RELATED APPLICATIONS

This application is a continuation, and claims benefit, of U.S.application Ser. No. 12/571,949, filed on Oct. 1, 2009, which claimspriority to U.S. Provisional Application Ser. No. 61/101,809, filed Oct.1, 2008. The contents of the prior applications are incorporated hereinby reference in their entirety.

This invention was made with government support under Grant No.DMR0213282 awarded by the National Science Foundation and under GrantNo. DE-FG02-99ER45778 awarded by the Department of Energy and underContract No. W911NF-07-D-0004 awarded by the Army Research Office. Thegovernment has certain rights in the invention.

BACKGROUND

The disclosure relates to wireless energy transfer. Wireless energytransfer can for example, be useful in such applications as providingpower to autonomous electrical or electronic devices.

Radiative modes of omni-directional antennas (which work very well forinformation transfer) are not suitable for such energy transfer, becausea vast majority of energy is wasted into free space. Directed radiationmodes, using lasers or highly-directional antennas, can be efficientlyused for energy transfer, even for long distances (transfer distanceL_(TRANS)>>L_(DEV), where L_(DEV) is the characteristic size of thedevice and/or the source), but may require existence of anuninterruptible line-of-sight and a complicated tracking system in thecase of mobile objects. Some transfer schemes rely on induction, but aretypically restricted to very close-range (L_(TRANS)<<L_(DEV)) or lowpower (˜mW) energy transfers.

The rapid development of autonomous electronics of recent years (e.g.laptops, cell-phones, house-hold robots, that all typically rely onchemical energy storage) has led to an increased need for wirelessenergy transfer.

SUMMARY

Disclosed is a method for transferring energy wirelessly. The methodincludes i) transferring energy wirelessly from a first resonatorstructure to an intermediate resonator structure, wherein the couplingrate between the first resonator structure and the intermediateresonator structure is κ_(1B); ii) transferring energy wirelessly fromthe intermediate resonator structure to a second resonator structure,wherein the coupling rate between the intermediate resonator structureand the second resonator structure is κ_(B2); and iii) during thewireless energy transfers, adjusting at least one of the coupling ratesκ_(1B) and κ_(B2) to reduce energy accumulation in the intermediateresonator structure and improve wireless energy transfer from the firstresonator structure to the second resonator structure through theintermediate resonator structure.

Embodiments of the method may include one or more of the followingfeatures.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to minimize energy accumulation in the intermediateresonator structure and cause wireless energy transfer from the firstresonator structure to the second resonator structure.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to maintain energy distribution in the field of thethree-resonator system in an eigenstate having substantially no energyin the intermediate resonator structure. For example, the adjustment ofat least one of the coupling rates κ_(1B) and κ_(B2) can further causethe eigenstate to evolve substantially adiabatically from an initialstate with substantially all energy in the resonator structures in thefirst resonator structure to a final state with substantially all of theenergy in the resonator structures in the second resonator structure.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to include adjustments of both coupling rates κ_(1B) andκ_(B2) during wireless energy transfer.

The resonator structures can each have a quality factor larger than 10.

The first and second resonator structures can each have a quality factorgreater than 50.

The first and second resonator structures can each have a quality factorgreater than 100.

The resonant energy in each of the resonator structures can includeelectromagnetic fields. For example, the maximum value of the couplingrate κ_(1B) and the maximum value of the coupling rate κ_(B2) forinductive coupling between the intermediate resonator structure and eachof the first and second resonator structures can each be larger thantwice the loss rate Γ for each of the first and second resonators.Moreover, The maximum value of the coupling rate κ_(1B) and the maximumvalue of the coupling rate κ_(B2) for inductive coupling between theintermediate resonator structure and each of the first and secondresonator structures can each be larger than four (4) times the lossrate Γ for each of the first and second resonators.

Each resonator structure can have a resonant frequency between 50 KHzand 500 MHz.

The maximum value of the coupling rate κ_(1B) and the maximum value ofthe coupling rate κ_(B2) can each be at least five (5) times greaterthan the coupling rate between the first resonator structure and thesecond resonator structure.

The intermediate resonator structure can have a rate of radiative energyloss that is at least twenty (20) times greater than that for either thefirst resonator structure or the second resonator structure.

The first and second resonator structures can be substantiallyidentical.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to cause peak energy accumulation in the intermediateresonator structure to be less than five percent (5%) of the peak totalenergy in the three resonator structures.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to cause peak energy accumulation in the intermediateresonator structure during the wireless energy transfers to be less thanten percent (10%) of the peak total energy in the three resonatorstructures.

Adjusting at least one of the coupling rates κ_(1B) and κ_(B2) caninclude adjusting a relative position and/or orientation between one ormore pairs of the resonator structures. Moreover, adjusting at least oneof the coupling rates κ_(1B) and κ_(B2) can include adjusting aresonator property of one or more of the resonator structures, such asmutual inductance.

The resonator structures can include a capacitively loaded loop or coilof at least one of a conducting wire, a conducting Litz wire, and aconducting ribbon.

The resonator structures can include an inductively loaded rod of atleast one of a conducting wire, a conducting Litz wire, and a conductingribbon.

The wireless energy transfers are non-radiative energy transfersmediated by a coupling of a resonant field evanescent tail of the firstresonator structure and a resonant field evanescent tail of theintermediate resonator structure and a coupling of the resonant fieldevanescent tail of the intermediate resonator structure and a resonantfield evanescent tail of the second resonator structure.

The adjustment of the at least one of the coupling rates can define afirst mode of operation, wherein the reduction in the energyaccumulation in the intermediate resonator structure is relative toenergy accumulation in the intermediate resonator structure for a secondmode of operation of wireless energy transfer among the three resonatorstructures having a coupling rate κ′_(1B) for wireless energy transferfrom the first resonator structure to the intermediate resonatorstructure and a coupling rate κ′_(B2) for wireless energy transfer fromthe intermediate resonator structure to the second resonator structurewith κ′_(1B) and κ′_(B2) each being substantially constant during thesecond mode of wireless energy transfer, and wherein the adjustment ofthe coupling rates κ_(1B) and κ_(2B) in the first mode of operation canbe selected to κ_(1B),κ_(B2)<√{square root over ((κ′_(1B) ²+κ′_(B2)²)/2)}. Moreover, the first mode of operation can have a greaterefficiency of energy transferred from the first resonator to the secondresonator compared to that for the second mode of operation. Further,the first and second resonator structures can be substantially identicaland each one can have a loss rate Γ_(A), the intermediate resonatorstructure can have a loss rate Γ_(B), and wherein Γ_(B)/Γ_(A) can begreater than 50.

Also, a ratio of energy lost to radiation and total energy wirelesslytransferred between the first and second resonator structures in thefirst mode of operation is less than that for the second mode ofoperation. Moreover, the first and second resonator structures can besubstantially identical and each one can have a loss rate Γ_(A) and aloss rate only due to radiation Γ_(A,rad), the intermediate resonatorstructure can have a loss rate Γ_(B) and a loss rate only due toradiation Γ_(B,rad) and wherein Γ_(B,rad)/Γ_(B)>Γ_(A,rad)/Γ_(A).

The first mode of operation the intermediate resonator structureinteracts less with extraneous objects than it does in the second modeof operation.

During the wireless energy transfer from the first resonator structureto the second resonator structure at least one of the coupling rates canbe adjusted so that κ_(1B)<<κ_(B2) at a start of the energy transfer andκ_(1B)>>κ_(B2) by a time a substantial portion of the energy has beentransferred from the first resonator structure to the second resonatorstructure.

The coupling rate κ_(B2) can be maintained at a fixed value and thecoupling rate κ_(1B) is increased during the wireless energy transferfrom the first resonator structure to second resonator structure.

The coupling rate κ_(1B) can be maintained at a fixed value and thecoupling rate κ_(B2) is decreased during the wireless energy transferfrom the first resonator structure to second resonator structure.

During the wireless energy transfer from the first resonator structureto second resonator structure, the coupling rate κ_(1B) can be increasedand the coupling rate κ_(B2) is decreased.

The method may further include features corresponding to those listedfor one or more of the apparatuses and methods described below.

In another aspect, disclosed is an apparatus including: first,intermediate, and second resonator structures, wherein a coupling ratebetween the first resonator structure and the intermediate resonatorstructure is κ_(1B) and a coupling rate between the intermediateresonator structure and the second resonator structure is κ_(B2); andmeans for adjusting at least one of the coupling rates κ_(1B) and κ_(B2)during wireless energy transfers among the resonator structures toreduce energy accumulation in the intermediate resonator structure andimprove wireless energy transfer from the first resonator structure tothe second resonator structure through the intermediate resonatorstructure.

Embodiments for the apparatus can include one or more of the followingfeatures.

The means for adjusting can include a rotation stage for adjusting therelative orientation of the intermediate resonator structure withrespect to the first and second resonator structures.

The means for adjusting can include a translation stage for moving thefirst and/or second resonator structures relative to the intermediateresonator structure.

The means for adjusting can include a mechanical, electro-mechanical, orelectrical staging system for dynamically adjusting the effective sizeof one or more of the resonator structures.

The apparatus may further include features corresponding to those listedfor the method described above, and one or more of the apparatuses andmethods described below.

In another aspect, a method for transferring energy wirelessly includesi): transferring energy wirelessly from a first resonator structure to aintermediate resonator structure, wherein the coupling rate between thefirst resonator structure and the intermediate resonator structure isκ_(1B); ii) transferring energy wirelessly from the intermediateresonator structure to a second resonator, wherein the coupling ratebetween the intermediate resonator structure and the second resonatorstructure is κ_(B2); and iii) during the wireless energy transfers,adjusting at least one of the coupling rates κ_(1B) and κ_(B2) to causean energy distribution in the field of the three-resonator system tohave substantially no energy in the intermediate resonator structurewhile wirelessly transferring energy from the first resonator structureto the second resonator structure through the intermediate resonatorstructure.

Embodiments for the method above can include one or more of thefollowing features.

Having substantially no energy in the intermediate resonator structurecan mean that peak energy accumulation in the intermediate resonatorstructure is less than ten percent (10%) of the peak total energy in thethree resonator structures throughout the wireless energy transfer.

Having substantially no energy in the intermediate resonator structurecan mean that peak energy accumulation in the intermediate resonatorstructure is less than five percent (5%) of the peak total energy in thethree resonator structures throughout the wireless energy transfer.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to maintain the energy distribution in the field of thethree-resonator system in an eigenstate having the substantially noenergy in the intermediate resonator structure.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to further cause the eigenstate to evolve substantiallyadiabatically from an initial state with substantially all energy in theresonator structures in the first resonator structure to a final statewith substantially all of the energy in the resonator structures in thesecond resonator structure.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can include adjustments of both coupling rates κ_(1B) and κ_(B2) duringwireless energy transfers.

The resonant energy in each of the resonator structures compriseselectromagnetic fields. For example, the maximum value of the couplingrate κ_(1B) and the maximum value of the coupling rate κ_(B2) forinductive coupling between the intermediate resonator structure and eachof the first and second resonator structures can each be larger thantwice the loss rate Γ for each of the first and second resonators.Moreover, the maximum value of the coupling rate κ_(1B) and the maximumvalue of the coupling rate κ_(B2) for inductive coupling between theintermediate resonator structure and each of the first and secondresonator structures can each be larger than four (4) times the lossrate Γ for each of the first and second resonators.

The resonator structure can have a resonant frequency between 50 KHz and500 MHz.

The maximum value of the coupling rate κ_(1B) and the maximum value ofthe coupling rate κ_(B2) can each be at least five (5) times greaterthan the coupling rate between the first resonator structure and thesecond resonator structure.

The intermediate resonator structure can have a rate of radiative energyloss that is at least twenty (20) times greater than that for either thefirst resonator structure or the second resonator structure.

The first and second resonator structures can be substantiallyidentical.

Adjusting at least one of the coupling rates κ_(1B) and κ_(B2) caninclude adjusting a relative position and/or orientation between one ormore pairs of the resonator structures.

Adjusting at least one of the coupling rates κ_(1B) and κ_(B2) caninclude adjusting a resonator property of one or more of the resonatorstructures, such as mutual inductance.

The resonator structures can include a capacitively loaded loop or coilof at least one of a conducting wire, a conducting Litz wire, and aconducting ribbon.

The resonator structures can include an inductively loaded rod of atleast one of a conducting wire, a conducting Litz wire, and a conductingribbon.

The wireless energy transfers can be non-radiative energy transfersmediated by a coupling of a resonant field evanescent tail of the firstresonator structure and a resonant field evanescent tail of theintermediate resonator structure and a coupling of the resonant fieldevanescent tail of the intermediate resonator structure and a resonantfield evanescent tail of the second resonator structure.

The first and second resonator structures can each have a quality factorgreater than 50.

The first and second resonator structures can each have a quality factorgreater than 100.

The adjustment of at least one of the coupling rates κ_(1B) and κ_(B2)can be selected to cause the energy distribution in the field of thethree-resonator system to have substantially no energy in theintermediate resonator structure improves wireless energy transferbetween the first and second resonator structures.

The adjustment of the at least one of the coupling rates can be selectedto define a first mode of operation, wherein energy accumulation in theintermediate resonator structure during the wireless energy transferfrom the first resonator structure to second resonator structure issmaller than that for a second mode of operation of wireless energytransfer among the three resonator structures having a coupling rateκ′_(1B) for wireless energy transfer from the first resonator structureto the intermediate resonator structure and a coupling rate ic κ′_(B2)for wireless energy transfer from the intermediate resonator structureto the second resonator structure with κ′_(1B) and κ′_(B2) each beingsubstantially constant during the second mode of wireless energytransfer, and wherein the adjustment of the coupling rates κ_(1B) andκ_(B2) in the first mode of operation can be selected to satisfyκ_(1B),κ_(B2)<√{square root over ((κ′_(1B) ²+κ′_(B2) ²)/2)}.

The first mode of operation can have a greater efficiency of energytransferred from the first resonator to the second resonator compared tothat for the second mode of operation.

The first and second resonator structures can be substantially identicaland each one can have a loss rate Γ_(A), the intermediate resonatorstructure can have a loss rate Γ_(B), and wherein Γ_(B)/Γ_(A) can begreater than 50.

A ratio of energy lost to radiation and total energy wirelesslytransferred between the first and second resonator structures in thefirst mode of operation can be less than that for the second mode ofoperation.

The first and second resonator structures can be substantially identicaland each one can have a loss rate Γ_(A) and a loss rate only due toradiation Γ_(A,rad), the intermediate resonator structure can have aloss rate Γ_(B) and a loss rate only due to radiation Γ_(B,rad) andwherein Γ_(B,rad)/Γ_(B)>Γ_(A,rad)/Γ_(A).

The first mode of operation the intermediate resonator structureinteracts less with extraneous objects than it does in the second modeof operation.

During the wireless energy transfer from the first resonator structureto the second resonator structure at least one of the coupling rates canbe adjusted so that κ_(1B)<<κ_(B2) at a start of the energy transfer andκ_(1B)>>κ_(B2) by a time a substantial portion of the energy has beentransferred from the first resonator structure to the second resonatorstructure.

The coupling rate κ_(B2) can be maintained at a fixed value and thecoupling rate κ_(1B) can be increased during the wireless energytransfer from the first resonator structure to second resonatorstructure.

The coupling rate κ_(1B) can be maintained at a fixed value and thecoupling rate κ_(B2) can be decreased during the wireless energytransfer from the first resonator structure to second resonatorstructure.

During the wireless energy transfer from the first resonator structureto second resonator structure, the coupling rate κ_(1B) can be increasedand the coupling rate κ_(B2) can be decreased.

The method may further include features corresponding to those listedfor the apparatus and method described above, and one or more of theapparatuses and methods described below.

In another aspect, disclosed is an apparatus including: first,intermediate, and second resonator structures, wherein a coupling ratebetween the first resonator structure and the intermediate resonatorstructure is κ_(1B) and a coupling rate between the intermediateresonator structure and the second resonator structure is κ_(B2); andmeans for adjusting at least one of the coupling rates κ_(1B) and κ_(B2)during wireless energy transfers among the resonator structures to causean energy distribution in the field of the three-resonator system tohave substantially no energy in the intermediate resonator structurewhile wirelessly transferring energy from the first resonator structureto the second resonator structure through the intermediate resonatorstructure.

Embodiments for the apparatus can include one or more of the followingfeatures.

Having substantially no energy in the intermediate resonator structurecan mean that peak energy accumulation in the intermediate resonatorstructure is less than ten percent (10%) of the peak total energy in thethree resonator structures throughout the wireless energy transfers.

Having substantially no energy in the intermediate resonator structurecan mean that peak energy accumulation in the intermediate resonatorstructure is less than five percent (5%) of the peak total energy in thethree resonator structures throughout the wireless energy transfers.

The means for adjusting can be configured to maintain the energydistribution in the field of the three-resonator system in an eigenstatehaving the substantially no energy in the intermediate resonatorstructure.

The means for adjusting can include a rotation stage for adjusting therelative orientation of the intermediate resonator structure withrespect to the first and second resonator structures.

The means for adjusting can include a translation stage for moving thefirst and/or second resonator structures relative to the intermediateresonator structure.

The means for adjusting can include a mechanical, electro-mechanical, orelectrical staging system for dynamically adjusting the effective sizeof one or more of the resonator structures.

The resonator structures can include a capacitively loaded loop or coilof at least one of a conducting wire, a conducting Litz wire, and aconducting ribbon.

The resonator structures can include an inductively loaded rod of atleast one of a conducting wire, a conducting Litz wire, and a conductingribbon.

A source can be coupled to the first resonator structure and a load canbe coupled to the second resonator structure.

The apparatus may further include features corresponding to those listedfor the apparatus and methods described above, and the apparatus andmethod described below.

In another aspect, disclosed is a method for transferring energywirelessly that includes: i) transferring energy wirelessly from a firstresonator structure to a intermediate resonator structure, wherein thecoupling rate between the first resonator structure and the intermediateresonator structure is κ_(1B); ii) transferring energy wirelessly fromthe intermediate resonator structure to a second resonator, wherein thecoupling rate between the intermediate resonator structure and thesecond resonator structure with a coupling rate is κ_(B2); and iii)during the wireless energy transfers, adjusting at least one of thecoupling rates κ_(1B) and κ_(B2) to define a first mode of operation inwhich energy accumulation in the intermediate resonator structure isreduced relative to that for a second mode of operation of wirelessenergy transfer among the three resonator structures having a couplingrate κ′_(1B) for wireless energy transfer from the first resonatorstructure to the intermediate resonator structure and a coupling rateκ′_(B2) for wireless energy transfer from the intermediate resonatorstructure to the second resonator structure with κ′_(1B) and κ′_(B2)each being substantially constant during the second mode of wirelessenergy transfer, and wherein the adjustment of the coupling rates κ_(1B)and κ_(B2) in the first mode of operation can be selected to satisfyκ_(1B),κ_(B2)<√{square root over ((κ′_(1B) ²+κ′_(B2) ²)/2)}.

The method may further include features corresponding to those listedfor the apparatuses and methods described above.

In another aspect, disclosed is an apparatus that includes: first,intermediate, and second resonator structures, wherein a coupling ratebetween the first resonator structure and the intermediate resonatorstructure is κ_(1B) and a coupling rate between the intermediateresonator structure and the second resonator structure is κ_(B2); andmeans for adjusting at least one of the coupling rates κ_(1B) and κ_(B2)during wireless energy transfers among the resonator structures todefine a first mode of operation in which energy accumulation in theintermediate resonator structure is reduced relative to that for asecond mode of operation for wireless energy transfer among the threeresonator structures having a coupling rate κ′_(1B) for wireless energytransfer from the first resonator structure to the intermediateresonator structure and a coupling rate κ′_(B2) for wireless energytransfer from the intermediate resonator structure to the secondresonator structure with κ′_(1B) and κ′_(B2) each being substantiallyconstant during the second mode of wireless energy transfer, and whereinthe adjustment of the coupling rates κ₁₂ and κ_(B2) in the first mode ofoperation can be selected to satisfy κ_(1B),κ_(B2)<√{square root over((κ′_(1B) ²+κ′_(B2) ²)/2)}.

The apparatus may further include features corresponding to those listedfor the apparatuses and methods described above.

Unless otherwise defined, all technical and scientific terms used hereinhave the same meaning as commonly understood by one of ordinary skill inthe art. Although methods and materials similar or equivalent to thosedescribed herein can be used in the practice or testing of the presentdisclosure, suitable methods and materials are described below. Allpublications, patent applications, patents, and other referencesmentioned herein are incorporated by reference in their entirety. Incase of conflict, the present specification, including definitions, willcontrol. In addition, the materials, methods, and examples areillustrative only and not intended to be limiting.

The details of one or more embodiments are set forth in the accompanyingdrawings and the description below, including the documents appendedhereto. Other features and advantages will be apparent from thisdisclosure and from the claims.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a schematic of an example wireless energy transfer scheme.

FIGS. 2( a)-(b) show the efficiency of power transmission η_(P) for (a)U=1 and (b) U=3, as a function of the frequency detuning D_(o) and fordifferent values of the loading rate U_(o).

FIG. 2( c) shows the optimal (for zero detuning and under conditions ofimpedance matching) efficiency for energy transfer η_(E*) and powertransmission η_(P*), as a function of the coupling-to-lossfigure-of-merit U.

FIG. 3 shows an example of a self-resonant conducting-wire coil.

FIG. 4 shows an example of a wireless energy transfer scheme featuringtwo self-resonant conducting-wire coils.

FIG. 5 is a schematic of an experimental system demonstrating wirelessenergy transfer.

FIG. 6 shows a comparison between experimental and theoretical resultsfor the coupling rate of the system shown schematically in FIG. 5.

FIG. 7 shows a comparison between experimental and theoretical resultsfor the strong-coupling factor of the system shown schematically in FIG.5.

FIG. 8 shows a comparison between experimental and theoretical resultsfor the power-transmission efficiency of the system shown schematicallyin FIG. 5.

FIG. 9 shows an example of a capacitively-loaded conducting-wire coil,and illustrates the surrounding field.

FIG. 10 shows an example wireless energy transfer scheme featuring twocapacitively-loaded conducting-wire coils, and illustrates thesurrounding field.

FIG. 11 illustrates an example circuit model for wireless energytransfer.

FIG. 12 shows the efficiency, total (loaded) device Q, and source anddevice currents, voltages and radiated powers (normalized to 1 Watt ofoutput power to the load) as functions of the resonant frequency, for aparticular choice of source and device loop dimensions, wp and N_(s) anddifferent choices of N_(d)=1, 2, 3, 4, 5, 6, 10.

FIG. 13 shows the efficiency, total (loaded) device Q, and source anddevice currents, voltages and radiated powers (normalized to 1 Watt ofoutput power to the load) as functions of frequency and wp for aparticular choice of source and device loop dimensions, and number ofturns N_(s) and N_(d).

FIG. 14 shows an example of an inductively-loaded conducting-wire coil.

FIG. 15 shows (a) an example of a resonant dielectric disk, andillustrates the surrounding field and (b) a wireless energy transferscheme featuring two resonant dielectric disks, and illustrates thesurrounding field.

FIG. 16 shows a schematic of an example wireless energy transfer schemewith one source resonator and one device resonator exchanging energyindirectly through an intermediate resonator.

FIG. 17 shows an example of a wireless energy transfer system: (a)(Left) Schematic of loops configuration in two-object direct transfer.(Right) Time evolution of energies in the two-object direct energytransfer case. (b) (Left) Schematic of three-loops configuration in theconstant-κ case. (Right) Dynamics of energy transfer for theconfiguration in (b. Left). Note that the total energy transferred E₂ is2 times larger than in (a. Right), but at the price of the total energyradiated being 4 times larger. (c) (Left) Loop configuration at t=0 inthe adiabatic-κ scheme. (Center) Dynamics of energy transfer withadiabatically rotating loops. (Right) Loop configuration at t=t_(EIT).Note that E₂ is comparable to (b. Right), but the radiated energy is nowmuch smaller: In fact, it is comparable to (a. Right).

FIG. 18 shows a schematic of an example wireless energy transfer schemewith one source resonator and one device resonator exchanging energyindirectly through an intermediate resonator, where an adjustment systemis used to rotate the resonator structures to dynamically adjust theircoupling rates.

FIG. 19 shows an example of a temporal variation of the coupling ratesin a wireless energy transfer system as in FIG. 18 to achieve anadiabatic transfer of energy from the source object R₁ to the deviceobject R₂.

FIG. 20 shows the energy distribution in a wireless energy transfersystem as in FIG. 18 as a function of time when the coupling rates aretime-varying, for Γ_(A)=0, κ/Γ_(B)=10, κ_(1B)=κ sin [πt/(2t_(EIT))], andκ_(B2)=κ cos [πt/(2t_(EIT))].

FIGS. 21( a)-(f) show a comparison between the adiabatic-κ andconstant-κ energy transfer schemes, in the general case: (a) Optimum E₂(%) in adiabatic-κ transfer, (b) Optimum E₂ (%) in constant-κ transfer,(c) (E₂)_(adiabatic-κ)/(E₂)_(constant-κ), (d) Energy lost (%) at optimumadiabatic-κ transfer, (e) Energy lost (%) at optimum constant-κtransfer, (f) (E_(lost))_(constant-κ)/(E_(lost))_(adiabatic-κ).

FIG. 22( a)-(e) show a comparison between radiated energies in theadiabatic-κ and constant-κ energy transfer schemes: (a) E_(rad) (%) inthe constant-scheme for Γ_(B)/Γ_(A)=500 and Γ_(rad) ^(A)=0, (b) E_(rad)(%) in the adiabatic-κ scheme for Γ_(B)/Γ_(A)=500 and Γ_(rad) ^(A)=0,(c) (E_(rad))_(constant-κ)/(E_(rad))_(adiabatic-κ) for Γ_(B)/Γ_(A)=50,(d) (E_(rad))_(constant-κ)/(E_(rad))_(adiabatic-κ) for Γ_(B)/Γ_(A)=500,(e) [(E_(rad))_(constant-κ)/(E_(rad))_(adiabatic-κ)] as function ofκ/Γ_(B) and Γ_(B)/Γ_(A), for Γ_(rad) ^(A)=0.

FIGS. 23( a)-(b) show schematics for frequency control mechanisms.

FIGS. 24( a)-(c) illustrate a wireless energy transfer scheme using twodielectric disks in the presence of various extraneous objects.

DETAILED DESCRIPTION

Efficient wireless energy-transfer between two similar-frequencyresonant objects can be achieved at mid-range distances, provided theseresonant objects are designed to operate in the ‘strong-coupling’regime. ‘Strong coupling’ can be realized for a wide variety of resonantobjects, including electromagnetic resonant objects such asinductively-loaded conducting rods and dielectric disks. Recently, wehave demonstrated wireless energy transfer between strongly coupledelectromagnetic self-resonant conducting coils and capacitively-loadedconducting coils, bearing high-Q electromagnetic resonant modes. See,for example, the following commonly owned U.S. patent applications, allof which are incorporated herein by reference: U.S. application Ser. No.11/481,077, filed on Jul. 5, 2006, and published as U.S. PatentPublication No. US 2007-0222542 A1; U.S. application Ser. No.12/055,963, filed on Mar. 26, 2008, and published as U.S. PatentPublication No. US 2008-0278264 A1; and U.S. patent application Ser. No.12/466,065, filed on May 14, 2009, and published as U.S. PatentPublication No. 20090284083. In general, the energy-transfer efficiencybetween similar-frequency, strongly coupled resonant objects decreasesas the distance between the objects is increased.

In this work, we explore a further scheme of efficient energy transferbetween resonant objects that extends the range over which energy may beefficiently transferred. Instead of transferring energy directly betweentwo resonant objects, as has been described in certain embodiments ofthe cross-referenced patents, in certain embodiments, an intermediateresonant object, with a resonant frequency equal or nearly-equal to thatof the two energy-exchanging resonant objects is used to mediate thetransfer. The intermediate resonant object may be chosen so that itcouples more strongly to each of the resonant objects involved in theenergy transfer than those two resonant objects couple to each other.One way to design such an intermediate resonator is to make it largerthan either of the resonant objects involved in the energy transfer.However, increasing the size of the intermediate resonant object maylower its quality factor, or Q, by increasing its radiation losses.Surprisingly enough, this new “indirect” energy transfer scheme may beshown to be very efficient and only weakly-radiative by introducing ameticulously chosen time variation of the resonator coupling rates.

The advantage of this method over the prior commonly owned wirelessenergy transfer techniques is that, in certain embodiments, it canenable energy to be transferred wirelessly between two objects with alarger efficiency and/or with a smaller radiation loss and/or with fewerinteractions with extraneous objects.

Accordingly, in certain embodiments, we disclose an efficient wirelessenergy transfer scheme between two similar resonant objects, stronglycoupled to an intermediate resonant object of substantially differentproperties, but with the same resonance frequency. The transfermechanism essentially makes use of the adiabatic evolution of aninstantaneous (so called ‘dark’) resonant state of the coupledthree-object system. Our analysis is based on temporal coupled modetheory (CMT), and is general. Of particular commercial interest is theapplication of this technique to strongly-coupled electromagneticresonators used for mid-range wireless energy transfer applications. Weshow that in certain parameter regimes of interest, this scheme can bemore efficient, and/or less radiative than other wireless energytransfer approaches.

While the technique described herein is primarily directed to tangibleresonator structures, the technique shares certain features with aquantum interference phenomenon known in the atomic physics community asElectromagnetically Induced Transparency (EIT). In EIT, three atomicstates participate. Two of them, which are non-lossy, are coupled to onethat has substantial losses. However, by meticulously controlling themutual couplings between the states, one can establish a coupled systemwhich is overall non-lossy. This phenomena has been demonstrated usingcarefully timed optical pulses, referred to as probe laser pulses andStokes laser pulses, to reduce the opacity of media with the appropriatecollection of atomic states. A closely related phenomenon known asStimulated Raman Adiabatic Passage (STIRAP) may take place in a similarsystem; namely, the probe and Stokes laser beams may be used to achievecomplete coherent population transfer between two molecular states of amedium. Hence, we may refer to the currently proposed scheme as the“EIT-like” energy transfer scheme.

In certain embodiments, we disclose an efficient near-field energytransfer scheme between two similar resonant objects, based on anEIT-like transfer of the energy through a mediating resonant object withthe same resonant frequency. In embodiments, this EIT-like energytransfer may be realized using electromagnetic resonators as have beendescribed in the cross-referenced patents, but the scheme is not boundonly to wireless energy transfer applications. Rather, this scheme isgeneral and may find applications in various other types of couplingbetween general resonant objects. In certain embodiments describedbelow, we describe particular examples of electromagnetic resonators,but the nature of the resonators and their coupling mechanisms could bequite different (e.g. acoustic, mechanical, etc.). To the extent thatmany resonant phenomena can be modeled with nearly identical CMTequations, similar behavior to that described herein would occur.

1. Efficient Energy-Transfer by Two ‘Strongly Coupled’ Resonances

FIG. 1 shows a schematic that generally describes one example of theinvention, in which energy is transferred wirelessly between tworesonant objects. Referring to FIG. 1, energy is transferred over adistance D, between a resonant source object having a characteristicsize r₁ and a resonant device object of characteristic size r₂. Bothobjects are resonant objects. The wireless near-field energy transfer isperformed using the field (e.g. the electromagnetic field or acousticfield) of the system of two resonant objects.

The characteristic size of an object can be regarded as being equal tothe radius of the smallest sphere which can fit around the entireobject. The characteristic thickness of an object can be regarded asbeing, when placed on a flat surface in any arbitrary configuration, thesmallest possible height of the highest point of the object above a flatsurface. The characteristic width of an object can be regarded as beingthe radius of the smallest possible circle that the object can passthrough while traveling in a straight line. For example, thecharacteristic width of a cylindrical object is the radius of thecylinder.

Initially, we present a theoretical framework for understandingnear-field wireless energy transfer. Note however that it is to beunderstood that the scope of the invention is not bound by theory.

Different temporal schemes can be employed, depending on theapplication, to transfer energy between two resonant objects. Here wewill consider two particularly simple but important schemes: a one-timefinite-amount energy-transfer scheme and a continuous finite-rateenergy-transfer (power) scheme.

1.1 Finite-Amount Energy-Transfer Efficiency

Let the source and device objects be 1, 2 respectively and theirresonance modes, which we will use for the energy exchange, have angularfrequencies ω_(1,2), frequency-widths due to intrinsic (absorption,radiation etc.) losses Γ_(1,2) and (generally) vector fields F_(1,2)(r),normalized to unity energy. Once the two resonant objects are brought inproximity, they can interact and an appropriate analytical framework formodeling this resonant interaction is that of the well-knowncoupled-mode theory (CMT). This model works well, when the resonancesare well defined by having large quality factors and their resonantfrequencies are relatively close to each other. In this picture, thefield of the system of the two resonant objects 1, 2 can be approximatedby F(r,t)=a₁(t)F₁(r)+a₂(t)F₂ (r), where a_(1,2)(t) are the fieldamplitudes, with |a_(1,2)(t)|² equal to the energy stored inside theobject 1, 2 respectively, due to the normalization. Then, using e^(−iωt)time dependence, the field amplitudes can be shown to satisfy, to lowestorder:

$\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}t}{a_{1}(t)}} = {{{- {{\mathbb{i}}\left( {\omega_{1} - {{\mathbb{i}}\;\Gamma_{1}}} \right)}}{a_{1}(t)}} + {{\mathbb{i}}\;\kappa_{11}{a_{1}(t)}} + {{\mathbb{i}}\;\kappa_{12}{a_{2}(t)}}}}{{\frac{\mathbb{d}}{\mathbb{d}t}{a_{2}(t)}} = {{{- {{\mathbb{i}}\left( {\omega_{2} - {{\mathbb{i}}\;\Gamma_{2}}} \right)}}{a_{2}(t)}} + {{\mathbb{i}}\;\kappa_{21}{a_{1}(t)}} + {{\mathbb{i}}\;\kappa_{22}{a_{2}(t)}}}}} & (1)\end{matrix}$where κ_(11,22) are the shifts in each object's frequency due to thepresence of the other, which are a second-order correction and can beabsorbed into the resonant frequencies (eigenfrequencies) by settingω_(1,2)→ω_(1,2)+κ_(11,22), and κ_(12,21) are the coupling coefficients,which from the reciprocity requirement of the system satisfy κ₂₁=κ₁₂≡κ.

The resonant modes of the combined system are found by substituting[a₁(t),a₂(t)]=[A₁,A₂]e^(−i ωt). They have complex resonant frequenciesω _(±)=ω₁₂±√{square root over ((Δω₁₂)₂+κ²)}  (2a)where ω₁₂=[(ω₁+ω₂)−i(Γ₁+Γ₂)]/2, Δω₁₂=[(ω₁−ω₂)−i(Γ₁−Γ₂)]/2 and whosesplitting we denote as δ_(E)≡ ω ₊− ω ⁻, and corresponding resonant fieldamplitudes

$\begin{matrix}{{\overset{->}{V} \pm \begin{bmatrix}A_{1} \\A_{2}\end{bmatrix}_{\pm}} = {\begin{bmatrix}\kappa \\{{\Delta\;\omega_{12}} \mp \sqrt{\left( {\Delta\;\omega_{12}} \right)^{2} + \kappa^{2}}}\end{bmatrix}.}} & \left( {2b} \right)\end{matrix}$Note that, at exact resonance ω₁=ω₂=ω_(A) and for Γ₁=Γ₂=Γ_(A), we getΔω₁₂=0, δ_(E)=2κ, and then

${\overset{\_}{\omega}}_{\pm} = {{\omega_{A} \pm \kappa} - {{\mathbb{i}}\;\Gamma_{A}}}$${{\overset{->}{V}}_{\pm} = {\begin{bmatrix}A_{1} \\A_{2}\end{bmatrix}_{\pm} = \begin{bmatrix}1 \\{\mp 1}\end{bmatrix}}},$namely we get the known result that the resonant modes split to a lowerfrequency even mode and a higher frequency odd mode.

Assume now that at time t=0 the source object 1 has finite energy|a₁(0)|², while the device object has |a₂(0)|²=0. Since the objects arecoupled, energy will be transferred from 1 to 2. With these initialconditions, Eqs. (1) can be solved, predicting the evolution of thedevice field-amplitude to be

$\begin{matrix}{\frac{a_{2}(t)}{{a_{1}(0)}} = {\frac{2\kappa}{\delta_{E}}{\sin\left( \frac{\delta_{E}t}{2} \right)}{\mathbb{e}}^{\frac{\Gamma_{1}\Gamma_{2}}{2}t}{{\mathbb{e}}^{{- {\mathbb{i}}}\;\frac{\omega_{1} + \omega_{2}}{2}t}.}}} & (3)\end{matrix}$The energy-transfer efficiency will be η_(E)≡|a₂(t)|²/|a₁(0)|². Theratio of energy converted to loss due to a specific loss mechanism inresonators 1 and 2, with respective loss rates Γ_(1,loss) and Γ_(2,loss)will be η_(loss,E)=∫₀^(t)dτ[2Γ_(1,loss)|a₁(τ)|²+2Γ_(2,loss)|a₂(τ)|²]/|a₁(0)|². Note that, atexact resonance ω₁=ω₂=ω_(A) (an optimal condition), Eq. (3) can bewritten as

$\begin{matrix}{\frac{a_{2}(T)}{{a_{1}(0)}} = {{\frac{\sin\left( {\sqrt{1 - \Delta^{2}}T} \right)}{\sqrt{1 - \Delta^{2}}} \cdot {\mathbb{e}}^{{- T}/U}}{\mathbb{e}}^{{- {\mathbb{i}\omega}_{A}}t}}} & (4)\end{matrix}$where ≡κt, Δ⁻¹=2κ/(Γ₂−Γ₁) and U=2κ/(Γ₁+Γ₂).

In some examples, the system designer can adjust the duration of thecoupling t at will. In some examples, the duration t can be adjusted tomaximize the device energy (and thus efficiency η_(E)). Then, it can beinferred from Eq. (4) that η_(E) is maximized for

$\begin{matrix}{T_{*} = \frac{\tan^{- 1}\left( {U\sqrt{1 - \Delta^{2}}} \right)}{\sqrt{1 - \Delta^{2}}}} & (5)\end{matrix}$resulting in an optimal energy-transfer efficiency

$\begin{matrix}{{\eta_{E*} \equiv {\eta_{E}\left( T_{*} \right)}} = {\frac{U^{2}}{1 + {U^{2}\left( {1 - \Delta^{2}} \right)}}{{\exp\left( {- \frac{2{\tan^{- 1}\left( {U\sqrt{1 - \Delta^{2}}} \right)}}{U\sqrt{1 - \Delta^{2}}}} \right)}.}}} & \left( {6a} \right)\end{matrix}$which is a monotonically increasing function of the coupling-to-lossratio U=2κ/(Γ₁+Γ₂) and tends to unity when U

1

|Δ|⁻¹>>1. Therefore, the energy transfer is nearly perfect, when thecoupling rate is much faster than all loss rates (κ/Γ_(1,2)

1). In FIG. 2( c) we show the optimal energy-transfer efficiency whenΓ₁=Γ₂=Γ_(A)

Δ=0:

$\begin{matrix}{{\eta_{E}\left( {T_{*},{\Delta = 0}} \right)} = {\frac{U^{2}}{1 + U^{2}}{{\exp\left( {- \frac{2\tan^{- 1}U}{U}} \right)}.}}} & \left( {6b} \right)\end{matrix}$

In a real wireless energy-transfer system, the source object can beconnected to a power generator (not shown in FIG. 1), and the deviceobject can be connected to a power consuming load (e.g. a resistor, abattery, an actual device, not shown in FIG. 1). The generator willsupply the energy to the source object, the energy will be transferredwirelessly and non-radiatively from the source object to the deviceobject, and the load will consume the energy from the device object. Toincorporate such supply and consumption mechanisms into this temporalscheme, in some examples, one can imagine that the generator is verybriefly but very strongly coupled to the source at time t=0 to almostinstantaneously provide the energy, and the load is similarly verybriefly but very strongly coupled to the device at the optimal timet=t_(*) to almost instantaneously drain the energy. For a constantpowering mechanism, at time t=t_(*) also the generator can again becoupled to the source to feed a new amount of energy, and this processcan be repeated periodically with a period t_(*).

1.2 Finite-Rate Energy-Transfer (Power-Transmission) Efficiency

Let the generator be continuously supplying energy to the source object1 at a rate κ₁ and the load continuously draining energy from the deviceobject 2 at a rate κ₂. Field amplitudes s_(±1,2)(t) are then defined, sothat |s_(±1,2)(t)|² is equal to the power ingoing to (for the + sign) oroutgoing from (for the − sign) the object 1, 2 respectively, and the CMTequations are modified to

$\begin{matrix}{{{\frac{\mathbb{d}}{\mathbb{d}t}{a_{1}(t)}} = {{{- {i\left( {\omega_{1} - {i\;\Gamma_{1}}} \right)}}{a_{1}(t)}} + {i\;\kappa_{11}{a_{1}(t)}} + {i\;\kappa_{12}{a_{2}(t)}} - {\kappa_{1}{a_{1}(t)}} + {\sqrt{2\kappa_{1}}{s_{+ 1}(t)}}}}\mspace{79mu}{{\frac{\mathbb{d}}{\mathbb{d}t}{a_{2}(t)}} = {{{- {i\left( {\omega_{2} - {i\;\Gamma_{2}}} \right)}}{a_{2}(t)}} + {i\;\kappa_{21}{a_{1}(t)}} + {i\;\kappa_{22}{a_{2}(t)}} - {\kappa_{2}{a_{2}(t)}}}}\mspace{79mu}{{s_{- 1}(t)} = {{\sqrt{2\kappa_{1}}{a_{1}(t)}} - {s_{+ 1}(t)}}}\mspace{79mu}{{s_{- 2}(t)} = {\sqrt{2\kappa_{2}}{a_{2}(t)}}}} & (7)\end{matrix}$where again we can set ω_(1,2)→ω_(1,2)+κ_(11,22) and κ₂₁=κ₁₂≡κ.

Assume now that the excitation is at a fixed frequency ω, namely has theform s₊₁(t)=S₊₁e^(−iωt). Then the response of the linear system will beat the same frequency, namely a_(1,2)(t)=A_(1,2)e^(−iωt) ands_(−1,2)(t)=S_(−1,2)e^(−iωt). By substituting these into Eqs. (7), usingδ_(1,2)≡ω−ω_(1,2), and solving the system, we find the field-amplitudetransmitted to the load (S₂₁ scattering-matrix element)

$\begin{matrix}\begin{matrix}{S_{21} \equiv \frac{S_{- 2}}{S_{+ 1}}} \\{= \frac{2i\;\kappa\sqrt{\kappa_{1}\kappa_{2}}}{{\left( {\Gamma_{1} + \kappa_{1} - {i\;\delta_{1}}} \right)\left( {\Gamma_{2} + \kappa_{2} - {i\;\delta_{2}}} \right)} + \kappa^{2}}} \\{= \frac{2i\; U\sqrt{U_{1}U_{2}}}{{\left( {1 + U_{1} - {i\; D_{1}}} \right)\left( {1 + U_{2} - {i\; D_{2}}} \right)} + U^{2}}}\end{matrix} & (8)\end{matrix}$and the field-amplitude reflected to the generator (S₁₁scattering-matrix element)

$\begin{matrix}\begin{matrix}{S_{11} \equiv \frac{S_{- 1}}{S_{+ 1}}} \\{= \frac{{\left( {\Gamma_{1} - \kappa_{1} - {i\;\delta_{1}}} \right)\left( {\Gamma_{2} + \kappa_{2} - {i\;\delta_{2}}} \right)} + \kappa^{2}}{{\left( {\Gamma_{1} + \kappa_{1} - {i\;\delta_{1}}} \right)\left( {\Gamma_{2} + \kappa_{2} - {i\;\delta_{2}}} \right)} + \kappa^{2}}} \\{= \frac{{\left( {1 - U_{1} - {i\; D_{1}}} \right)\left( {1 + U_{2} - {i\; D_{2}}} \right)} + U^{2}}{{\left( {1 + U_{1} - {i\; D_{1}}} \right)\left( {1 + U_{2} - {i\; D_{2}}} \right)} + U^{2}}}\end{matrix} & (9)\end{matrix}$where D_(1,2)≡δ_(1,2)/Γ_(1,2), U_(1,2)≡κ_(1,2)/Γ_(1,2) and U≡κ/√{squareroot over (Γ₁Γ₂)}. Similarly, the scattering-matrix elements S₁₂, S₂₂are given by interchanging 1

2 in Eqs. (8), (9) and, as expected from reciprocity, S₂₁=S₁₂. Thecoefficients for power transmission (efficiency) and reflection and lossare respectively η_(P)≡|S₂₁|²=|S⁻²|²/|S₊₁|² and |S₁₁|²=|S⁻¹|²/|S₊₁|² and1−|S₂₁|²−|S₁₁|²=(2Γ₁|A₁|²+2Γ₂|A₂|²)/|S₊₁|².

In some implementations, the parameters D_(1,2), U_(1,2) can be designed(engineered), since one can adjust the resonant frequencies ω_(1,2)(compared to the desired operating frequency ω) and the generator/loadsupply/drain rates κ_(1,2). Their choice can target the optimization ofsome system performance-characteristic of interest.

In some examples, a goal can be to maximize the power transmission(efficiency) η_(P)≡|S₂₁|² of the system, so one would requireη_(P)′(D _(1,2))=η_(P)′(U _(1,2))=0  (10)

Since S₂₁ (from Eq. (8)) is symmetric upon interchanging 1

2, the optimal values for D_(1,2) (determined by Eqs. (10)) will beequal, namely D₁=D₂≡D_(o), and similarly U₁=U₂≡U_(o). Then,

$\begin{matrix}{S_{21} = \frac{2i\;{UU}_{o}}{\left( {1 + U_{o} - {i\; D_{o}}} \right)^{2} + U^{2}}} & (11)\end{matrix}$and from the condition η_(P)′(D_(o))=0 we get that, for fixed values ofU and U_(o), the efficiency can be maximized for the following values ofthe symmetric detuning

$\begin{matrix}{{D_{o} = \begin{matrix}{{\pm \sqrt{U^{2} - \left( {1 + U_{o}} \right)^{2}}},} & {{{if}\mspace{14mu} U} > {1 + U_{o}}} \\{0,} & {{{if}\mspace{14mu} U} \leq {1 + U_{o}}}\end{matrix}},} & (12)\end{matrix}$which, in the case U>1+U_(o), can be rewritten for the two frequenciesat which the efficiency peaks as

$\begin{matrix}{{\omega_{\pm} = {\frac{{\omega_{1}\Gamma_{2}} + {\omega_{2}\Gamma_{1}}}{\Gamma_{1} + \Gamma_{2}} \pm {\frac{2\sqrt{\Gamma_{1}\Gamma_{2}}}{\Gamma_{1} + \Gamma_{2}}\sqrt{\kappa^{2} - {\left( {\Gamma_{1} + \kappa_{1}} \right)\left( {\Gamma_{2} + \kappa_{2}} \right)}}}}},} & (13)\end{matrix}$whose splitting we denote as δ_(P)≡ ω ₊− ω ⁻. Note that, at exactresonance ω₁=ω₂, and for Γ₁=Γ₂≡Γ_(o) and κ₁=κ₂≡κ_(o), we getδ_(P)=2√{square root over (κ²−(Γ_(o)+κ_(o))²)}<δ_(E), namely thetransmission-peak splitting is smaller than the normal-mode splitting.Then, by substituting D_(o) into η_(P) from Eq. (12), from the conditionη_(P)′(U_(o))=0 we get that, for fixed value of U, the efficiency can bemaximized for

$\begin{matrix}{U_{o^{*}} = {{\sqrt{1 + U^{2}}\overset{{Eq}.{(12)}}{\Rightarrow}D_{o^{*}}} = 0}} & (14)\end{matrix}$which is known as ‘critical coupling’ condition, whereas forU_(o)<U_(o*) the system is called ‘undercoupled’ and for U_(o)>U_(o*) itis called ‘overcoupled’. The dependence of the efficiency on thefrequency detuning D_(o) for different values of U_(o) (including the‘critical-coupling’ condition) are shown in FIG. 2( a,b). The overalloptimal power efficiency using Eqs. (14) is

$\begin{matrix}{{{\eta_{P*} \equiv {\eta_{P}\left( {D_{o^{*}},U_{o^{*}}} \right)}} = {\frac{U_{o^{*}} - 1}{U_{o^{*}} + 1} = \left( \frac{U}{1 + \sqrt{1 + U^{2}}} \right)^{2}}},} & (15)\end{matrix}$which is again only a function of the coupling-to-loss ratioU=κ/√{square root over (Γ₁Γ₂)} and tends to unity when U

1, as depicted in FIG. 2( c).

In some examples, a goal can be to minimize the power reflection at theside of the generator |S₁₁|² and the load |S₂₂|², so one would then needS _(11,22)=0

(1∓U ₁ −iD ₁)(1±U ₂ −iD ₂)+U ²=0,  (16)

The equations above present ‘impedance matching’ conditions. Again, theset of these conditions is symmetric upon interchanging 1

2, so, by substituting D₁=D₂≡D_(o) and U₁=U₂≡U_(o) into Eqs. (16), weget(1−iD _(o))² −U _(o) ² +U ²=0,  (17)from which we easily find that the values of D_(o) and U_(o) that cancelall reflections are again exactly those in Eqs. (14).

It can be seen that, the two goals and their associated sets ofconditions (Eqs. (10) and Eqs. (16)) result in the same optimized valuesof the intra-source and intra-device parameters D_(1,2), U_(1,2). Notethat for a lossless system this would be an immediate consequence ofpower conservation (Hermiticity of the scattering matrix), but this isnot apparent for a lossy system.

Accordingly, for any temporal energy-transfer scheme, once theparameters specific only to the source or to the device (such as theirresonant frequencies and their excitation or loading rates respectively)have been optimally designed, the efficiency monotonically increaseswith the ratio of the source-device coupling-rate to their loss rates.Using the definition of a resonance quality factor Q=ω/2Γ and definingby analogy the coupling factor k≡1/Q_(κ)≡2κ/√{square root over (ω₁ω₂)},it is therefore exactly this ratio

$\begin{matrix}{U = {\frac{\kappa}{\sqrt{\Gamma_{1}\Gamma_{2}}} = {k\sqrt{Q_{1}Q_{2}}}}} & (18)\end{matrix}$that has been set as a figure-of-merit for any system underconsideration for wireless energy-transfer, along with the distance overwhich this ratio can be achieved (clearly, U will be a decreasingfunction of distance). The operating regime U>1 is sometimes called‘strong-coupling’ regime and is a sufficient condition for efficientenergy-transfer. In particular, for U>1 we get, from Eq. (15),η_(P*)>17%, large enough for many practical applications. Note that insome applications, U>0.1 may be sufficient. In applications where it isimpossible or impractical to run wires to supply power to a device,U<0.1 may be considered sufficient. One skilled in the art willrecognize that the sufficient U is application and specificationdependent. The figure-of-merit U may be called the strong-couplingfactor. We will further show how to design systems with a largestrong-coupling factor.

To achieve a large strong-coupling factor U, in some examples, theenergy-transfer application preferably uses resonant modes of highquality factors Q, corresponding to low (i.e. slow) intrinsic-loss ratesΓ. This condition can be satisfied by designing resonant modes where allloss mechanisms, typically radiation and absorption, are sufficientlysuppressed.

This suggests that the coupling be implemented using, not the lossyradiative far-field, which should rather be suppressed, but theevanescent (non-lossy) stationary near-field. To implement anenergy-transfer scheme, usually more appropriate are finite objects,namely ones that are topologically surrounded everywhere by air, intowhere the near field extends to achieve the coupling. Objects of finiteextent do not generally support electromagnetic states that areexponentially decaying in all directions in air away from the objects,since Maxwell's Equations in free space imply that k²=ω²/c², where k isthe wave vector, ω the angular frequency, and c the speed of light,because of which one can show that such finite objects cannot supportstates of infinite Q, rather there always is some amount of radiation.However, very long-lived (so-called “high-Q”) states can be found, whosetails display the needed exponential or exponential-like decay away fromthe resonant object over long enough distances before they turnoscillatory (radiative). The limiting surface, where this change in thefield behavior happens, is called the “radiation caustic”, and, for thewireless energy-transfer scheme to be based on the near field ratherthan the far/radiation field, the distance between the coupled objectsmust be such that one lies within the radiation caustic of the other.One typical way of achieving a high radiation-Q (Q_(rad)) is to designsubwavelength resonant objects. When the size of an object is muchsmaller than the wavelength of radiation in free space, itselectromagnetic field couples to radiation very weakly. Since the extentof the near-field into the area surrounding a finite-sized resonantobject is set typically by the wavelength, in some examples, resonantobjects of subwavelength size have significantly longer evanescentfield-tails. In other words, the radiation caustic is pushed far awayfrom the object, so the electromagnetic mode enters the radiative regimeonly with a small amplitude.

Moreover, most realistic materials exhibit some nonzero amount ofabsorption, which can be frequency dependent, and thus cannot supportstates of infinite Q, rather there always is some amount of absorption.However, very long-lived (“high-Q”) states can be found, whereelectromagnetic modal energy is only weakly dissipated. Some typicalways of achieving a high absorption-Q (Q_(abs)) is to use materialswhich exhibit very small absorption at the resonant frequency and/or toshape the field to be localized more inside the least lossy materials.

Furthermore, to achieve a large strong-coupling factor U, in someexamples, the energy-transfer application may use systems that achieve ahigh coupling factor k, corresponding to strong (i.e. fast) couplingrate K, over distances larger than the characteristic sizes of theobjects.

Since finite-sized subwavelength resonant objects can often be designedto have high Q, as was discussed above and will be seen in exampleslater on, such objects may typically be chosen for the resonantdevice-object. In these cases, the electromagnetic field is, in someexamples, of a quasi-static nature and the distance, up to whichsufficient coupling can be achieved, is dictated by the decay-law ofthis quasi-static field.

Note that in some examples, the resonant source-object may be immobileand thus less restricted in its allowed geometry and size. It can betherefore chosen to be large enough that the near-field extent is notlimited by the wavelength, and can thus have nearly infiniteradiation-Q. Some objects of nearly infinite extent, such as dielectricwaveguides, can support guided modes, whose evanescent tails aredecaying exponentially in the direction away from the object, slowly iftuned close to cutoff, therefore a good coupling can also be achievedover distances quite a few times larger than a characteristic size ofthe source- and/or device-object.

2 ‘Strongly-Coupled’ Resonances at Mid-Range Distances for RealisticSystems

In the following, examples of systems suitable for energy transfer ofthe type described above are described. We will demonstrate how tocompute the CMT parameters ω_(1,2), Q_(1,2) and k described above andhow to choose or design these parameters for particular examples inorder to produce a desirable figure-of-merit U=κ/√{square root over(Γ₁Γ₂)}=k√{square root over (Q₁Q₂)} at a desired distance D. In someexamples, this figure-of-merit is maximized when ω_(1,2) are tuned closeto a particular angular frequency ω_(U).

2.1 Self-Resonant Conducting Coils

In some examples, one or more of the resonant objects are self-resonantconducting coils. Referring to FIG. 3, a conducting wire of length, l,and cross-sectional radius, a, is wound into a helical coil of radius,r, and height, h, (namely with N=√{square root over (l²−h²)}/2πr numberof turns), surrounded by air. As described below, the wire hasdistributed inductance and distributed capacitance, and therefore itsupports a resonant mode of angular frequency ω. The nature of theresonance lies in the periodic exchange of energy from the electricfield within the capacitance of the coil, due to the charge distributionρ(x) across it, to the magnetic field in free space, due to the currentdistribution j(x) in the wire. In particular, the charge conservationequation ∇·j=iωρ implies that: (i) this periodic exchange is accompaniedby a π/2 phase-shift between the current and the charge densityprofiles, namely the energy W contained in the coil is at certain pointsin time completely due to the current and at other points in timecompletely due to the charge, and (ii) if ρ_(l)(x) and I(x) arerespectively the linear charge and current densities in the wire, wherex runs along the wire,

$q_{o} = {\frac{1}{2}{\int{{\mathbb{d}x}{{\rho_{l}(x)}}}}}$is the maximum amount of positive charge accumulated in one side of thecoil (where an equal amount of negative charge always also accumulatesin the other side to make the system neutral) and I_(o)=max{|I(x)|} isthe maximum positive value of the linear current distribution, thenI_(o)=ωq_(o). Then, one can define an effective total inductance L andan effective total capacitance C of the coil through the amount ofenergy W inside its resonant mode:

$\begin{matrix}{{\left. {W \equiv {\frac{1}{2}I_{o}^{2}L}}\Rightarrow L \right. = {\frac{\mu_{o}}{4\pi\; I_{o}^{2}}{\int{\int{{\mathbb{d}x}{\mathbb{d}x^{\prime}}\frac{{j(x)} \cdot {j\left( x^{\prime} \right)}}{{x - x^{\prime}}}}}}}},} & (19) \\{{\left. {W \equiv {\frac{1}{2}q_{o}^{2}\frac{1}{C}}}\Rightarrow\frac{1}{C} \right. = {\frac{1}{4{\pi ɛ}_{o}q_{o}^{2}}{\int{\int{{\mathbb{d}x}{\mathbb{d}x^{\prime}}\frac{\rho(x){\rho\left( x^{\prime} \right)}}{{x - x^{\prime}}}}}}}},} & (20)\end{matrix}$where μ_(o) and ∈_(o) are the magnetic permeability and electricpermittivity of free space.

With these definitions, the resonant angular frequency and the effectiveimpedance can be given by the formulas ω=1/√{square root over (LC)} andZ=√{square root over (L/C)} respectively.

Losses in this resonant system consist of ohmic (material absorption)loss inside the wire and radiative loss into free space. One can againdefine a total absorption resistance R_(abs) from the amount of powerabsorbed inside the wire and a total radiation resistance R_(rad) fromthe amount of power radiated due to electric- and magnetic-dipoleradiation:

$\begin{matrix}\left. {P_{abs} \equiv {\frac{1}{2}I_{o}^{2}R_{abs}}}\Rightarrow{R_{abs} \approx {\zeta_{c}{\frac{l}{2\pi\; a} \cdot \frac{I_{rms}^{2}}{I_{o}^{2}}}}} \right. & (21) \\{\left. {P_{rad} \equiv {\frac{1}{2}I_{o}^{2}R_{rad}}}\Rightarrow{R_{rad} \approx {\frac{\zeta_{o}}{6\pi}\left\lbrack {\left( \frac{\omega{p}}{c} \right)^{2} + \left( \frac{\omega\sqrt{m}}{c} \right)^{4}} \right\rbrack}} \right.,} & (22)\end{matrix}$where c=1/√{square root over (μ_(o)∈_(o))} and ζ_(o)=√{square root over(μ_(o)/∈_(o))} are the light velocity and light impedance in free space,the impedance ζ_(c) is ζ_(c)=1/σδ=√{square root over (μ_(o)ω/2σ)} with σthe conductivity of the conductor and δ the skin depth at the frequencyω,

${I_{rms}^{2} = {\frac{1}{l}{\int\;{{\mathbb{d}x}{{I(x)}}^{2}}}}},{p = {\int\;{{\mathbb{d}x}\mspace{11mu}{r_{\rho\; l}(x)}}}}$is the electric-dipole moment of the coil and

$m = {\frac{1}{2}{\int{{\mathbb{d}x}\mspace{11mu} r \times {j(x)}}}}$is the magnetic-dipole moment of the coil. For the radiation resistanceformula Eq. (22), the assumption of operation in the quasi-static regime(h,r

λ=2πc/ω) has been used, which is the desired regime of a subwavelengthresonance. With these definitions, the absorption and radiation qualityfactors of the resonance may be given by Q_(abs)=Z/R_(abs) andQ_(rad)=Z/R_(rad) respectively.

From Eq. (19)-(22) it follows that to determine the resonance parametersone simply needs to know the current distribution j in the resonantcoil. Solving Maxwell's equations to rigorously find the currentdistribution of the resonant electromagnetic eigenmode of aconducting-wire coil is more involved than, for example, of a standardLC circuit, and we can find no exact solutions in the literature forcoils of finite length, making an exact solution difficult. One could inprinciple write down an elaborate transmission-line-like model, andsolve it by brute force. We instead present a model that is (asdescribed below) in good agreement (˜5%) with experiment. Observing thatthe finite extent of the conductor forming each coil imposes theboundary condition that the current has to be zero at the ends of thecoil, since no current can leave the wire, we assume that the resonantmode of each coil is well approximated by a sinusoidal current profilealong the length of the conducting wire. We shall be interested in thelowest mode, so if we denote by x the coordinate along the conductor,such that it runs from −l/2 to +l/2, then the current amplitude profilewould have the form I(x)=I_(o) cos(πx/l), where we have assumed that thecurrent does not vary significantly along the wire circumference for aparticular x, a valid assumption provided a

r. It immediately follows from the continuity equation for charge thatthe linear charge density profile should be of the form ρ_(l)(x)=ρ_(o)sin(πx/l), and thus q_(o)=∫₀ ^(l/2)dxρ_(o)|sin(πx/l)|=ρ_(o)l/π. Usingthese sinusoidal profiles we find the so-called “self-inductance” L_(s)and “self-capacitance” C_(s) of the coil by computing numerically theintegrals Eq. (19) and (20); the associated frequency and effectiveimpedance are ω_(s) and Z_(s) respectively. The “self-resistances” R_(s)are given analytically by Eq. (21) and (22) using

$\mspace{79mu}{{I_{rms}^{2} = {{\frac{1}{l}{\int_{{- l}/2}^{l/2}\ {{\mathbb{d}x}{{I_{o}\cos\mspace{11mu}\left( {\pi\; x\text{/}l} \right)}}^{2}}}} = {\frac{1}{2}I_{o}^{2}}}},\mspace{79mu}{{p} = {q_{o}\sqrt{\left( {\frac{2}{\pi}h} \right)^{2} + \left( {\frac{4N\;\cos\;\left( {\pi\; N} \right)}{\left( {{4N^{2}} - 1} \right)\pi}r} \right)^{2}}}}}\mspace{14mu}$     and${{m} = {I_{o}\sqrt{\left( {\frac{2}{\pi}N\;\pi\; r^{2}} \right)^{2} + \left( {\frac{{{\cos\left( {\pi\; N} \right)}\left( {{12N^{2}} - 1} \right)} - {{\sin\left( {\pi\; N} \right)}\pi\;{N\left( {{4N^{2}} - 1} \right)}}}{\left( {{16N^{4}} - {8N^{2}} + 1} \right)\pi}{hr}} \right)^{2}}}},$and therefore the associated Q_(s) factors can be calculated.

The results for two examples of resonant coils with subwavelength modesof λ_(s)/r≧70 (i.e. those highly suitable for near-field coupling andwell within the quasi-static limit) are presented in Table 1. Numericalresults are shown for the wavelength and absorption, radiation and totalloss rates, for the two different cases of subwavelength-coil resonantmodes. Note that, for conducting material, copper (σ=5.998·10^−7 S/m)was used. It can be seen that expected quality factors at microwavefrequencies are Q_(s,abs)≧1000 and Q_(s,rad)≧5000.

TABLE 1 single coil λ_(s) /r f (MHz) Q_(s,rad) Q_(s,abs) Q_(s) r = 30cm, h = 20 cm, a = 1 cm, N = 4 74.7 13.39 4164 8170 2758 r = 10 cm, h =3 cm, a = 2 mm, N = 6 140 21.38 43919 3968 3639

Referring to FIG. 4, in some examples, energy is transferred between twoself-resonant conducting-wire coils. The electric and magnetic fieldsare used to couple the different resonant conducting-wire coils at adistance D between their centers. Usually, the electric coupling highlydominates over the magnetic coupling in the system under considerationfor coils with h

2r and, oppositely, the magnetic coupling highly dominates over theelectric coupling for coils with h

2r. Defining the charge and current distributions of two coils 1,2respectively as ρ_(1,2)(x) and j_(1,2)(x), total charges and peakcurrents respectively as q_(1,2) and I_(1,2), and capacitances andinductances respectively as C_(1,2) and L_(1,2), which are the analogsof ρ(x), j(x), q_(o), I_(o), C and L for the single-coil case and aretherefore well defined, we can define their mutual capacitance andinductance through the total energy:

$\begin{matrix}{{\left. {W \equiv {W_{1} + W_{2} + {\frac{1}{2}\left( {{q_{1}^{*}q_{2}} + {q_{2}^{*}q_{1}}} \right)\text{/}M_{C}} + {\frac{1}{2}\left( {{I_{1}^{*}I_{2}} + {I_{2}^{*}I_{1}}} \right)M_{L}}}}\Rightarrow{1/M_{C}} \right. = {\frac{1}{4{\pi ɛ}_{o\;}q_{1}q_{2}}{\int{\int{{\mathbb{d}x}{\mathbb{d}x^{\prime}}\frac{\rho_{1}(x){\rho_{2}\left( x^{\prime} \right)}}{{x - x^{\prime}}}u}}}}},\mspace{79mu}{M_{L} = {\frac{\mu_{o}}{4{\pi I}_{1}I_{2}}{\int{\int{{\mathbb{d}x}{\mathbb{d}x^{\prime}}\frac{{j_{1}(x)} \cdot {j_{2}\left( x^{\prime} \right)}}{{x - x^{\prime}}}u}}}}},} & (23)\end{matrix}$where

${W_{1} = {{\frac{1}{2}q_{1}^{2}\text{/}C_{1}} = {\frac{1}{2}I_{1}^{2}L_{1}}}},{W_{2} = {{\frac{1}{2}q_{2}^{2}\text{/}C_{2}} = {\frac{1}{2}I_{2}^{2}L_{2}}}}$and the retardation factor of u=exp(iω|x−x′|/c) inside the integral canbeen ignored in the quasi-static regime D

2 of interest, where each coil is within the near field of the other.With this definition, the coupling factor is given by k=√{square rootover (C₁C₂)}/M_(C)+M_(L)/√{square root over (L₁L₂)}.

Therefore, to calculate the coupling rate between two self-resonantcoils, again the current profiles are needed and, by using again theassumed sinusoidal current profiles, we compute numerically from Eq.(23) the mutual capacitance M_(C,s) and inductance M_(L,s) between twoself-resonant coils at a distance D between their centers, and thusk=1/Q_(κ) is also determined.

TABLE 2 pair of coils D/r Q Q_(κ) = 1/k U r = 30 cm, h = 20 cm, 3 275838.9 70.9 a = 1 cm, N = 4 5 2758 139.4 19.8 λ/r ≈ 75 7 2758 333.0 8.3Q_(s) ^(abs) ≈ 8170, Q_(s) ^(rad) ≈ 4164 10 2758 818.9 3.4 r = 10 cm, h= 3 cm, 3 3639 61.4 59.3 a = 2 mm, N = 6 5 3639 232.5 15.7 λ/r ≈ 140 73639 587.5 6.2 Q_(s) ^(abs) ≈ 3968, Q_(s) ^(rad) ≈ 43919 10 3639 15802.3

Referring to Table 2, relevant parameters are shown for exemplaryexamples featuring pairs or identical self resonant coils. Numericalresults are presented for the average wavelength and loss rates of thetwo normal modes (individual values not shown), and also the couplingrate and figure-of-merit as a function of the coupling distance D, forthe two cases of modes presented in Table 1. It can be seen that formedium distances D/r=10−3 the expected coupling-to-loss ratios are inthe range U˜2˜70.

2.1.1 Experimental Results

An experimental realization of an example of the above described systemfor wireless energy transfer consists of two self-resonant coils, one ofwhich (the source coil) is coupled inductively to an oscillatingcircuit, and the second (the device coil) is coupled inductively to aresistive load, as shown schematically in FIG. 5. Referring to FIG. 5, Ais a single copper loop of radius 25 cm that is part of the drivingcircuit, which outputs a sine wave with frequency 9.9 MHz. s and d arerespectively the source and device coils referred to in the text. B is aloop of wire attached to the load (“light-bulb”). The various κ'srepresent direct couplings between the objects. The angle between coil dand the loop A is adjusted so that their direct coupling is zero, whilecoils s and d are aligned coaxially. The direct coupling between B and Aand between B and s is negligible.

The parameters for the two identical helical coils built for theexperimental validation of the power transfer scheme were h=20 cm, a=3mm, r=30 cm and N=5.25. Both coils are made of copper. Due toimperfections in the construction, the spacing between loops of thehelix is not uniform, and we have encapsulated the uncertainty abouttheir uniformity by attributing a 10% (2 cm) uncertainty to h. Theexpected resonant frequency given these dimensions is f₀=10.56±0.3 MHz,which is approximately 5% off from the measured resonance at around 9.90MHz.

The theoretical Q for the loops is estimated to be ˜2500 (assumingperfect copper of resistivity ρ=1/σ=1.7×10⁻⁸ Ωm) but the measured valueis 950±50. We believe the discrepancy is mostly due to the effect of thelayer of poorly conducting copper oxide on the surface of the copperwire, to which the current is confined by the short skin depth (˜20 μm)at this frequency. We have therefore used the experimentally observed Q(and Γ₁=Γ₂=Γ=ω/(2Q) derived from it) in all subsequent computations.

The coupling coefficient κ can be found experimentally by placing thetwo self-resonant coils (fine-tuned, by slightly adjusting h, to thesame resonant frequency when isolated) a distance D apart and measuringthe splitting in the frequencies of the two resonant modes in thetransmission spectrum. According to Eq. (13) derived by coupled-modetheory, the splitting in the transmission spectrum should beδ_(P)=2√{square root over (κ²−δ²)}, when κ_(A,B) are kept very small bykeeping A and B at a relatively large distance. The comparison betweenexperimental and theoretical results as a function of distance when thetwo the coils are aligned coaxially is shown in FIG. 6.

FIG. 7 shows a comparison of experimental and theoretical values for thestrong-coupling factor U=κ/Γ as a function of the separation between thetwo coils. The theory values are obtained by using the theoreticallyobtained κ and the experimentally measured Γ. The shaded area representsthe spread in the theoretical U due to the ˜5% uncertainty in Q. Asnoted above, the maximum theoretical efficiency depends only on theparameter U, which is plotted as a function of distance in FIG. 7. U isgreater than 1 even for D=2.4 m (eight times the radius of the coils),thus the system is in the strongly-coupled regime throughout the entirerange of distances probed.

The power-generator circuit was a standard Colpitts oscillator coupledinductively to the source coil by means of a single loop of copper wire25 cm in radius (see FIG. 5). The load consisted of a previouslycalibrated light-bulb, and was attached to its own loop of insulatedwire, which was in turn placed in proximity of the device coil andinductively coupled to it. Thus, by varying the distance between thelight-bulb and the device coil, the parameter U_(B)=κ_(B)/Γ was adjustedso that it matched its optimal value, given theoretically by Eq. (14) asU_(B*)=√{square root over (1+U²)}. Because of its inductive nature, theloop connected to the light-bulb added a small reactive component toκ_(B) which was compensated for by slightly retuning the coil. The workextracted was determined by adjusting the power going into the Colpittsoscillator until the light-bulb at the load was at its full nominalbrightness.

In order to isolate the efficiency of the transfer taking placespecifically between the source coil and the load, we measured thecurrent at the mid-point of each of the self-resonant coils with acurrent-probe (which was not found to lower the Q of the coilsnoticeably.) This gave a measurement of the current parameters I₁ and I₂defined above. The power dissipated in each coil was then computed fromP_(1,2)=ΓL|I_(1,2)|², and the efficiency was directly obtained fromη=P_(B)/(P₁+P₂+P_(B)). To ensure that the experimental setup was welldescribed by a two-object coupled-mode theory model, we positioned thedevice coil such that its direct coupling to the copper loop attached tothe Colpitts oscillator was zero. The experimental results are shown inFIG. 8, along with the theoretical prediction for maximum efficiency,given by Eq. (15).

Using this example, we were able to transmit significant amounts ofpower using this setup from the source coil to the device coil, fullylighting up a 60 W light-bulb from distances more than 2 m away, forexample. As an additional test, we also measured the total power goinginto the driving circuit. The efficiency of the wirelesspower-transmission itself was hard to estimate in this way, however, asthe efficiency of the Colpitts oscillator itself is not precisely known,although it is expected to be far from 100%. Nevertheless, this gave anoverly conservative lower bound on the efficiency. When transmitting 60W to the load over a distance of 2 m, for example, the power flowinginto the driving circuit was 400 W. This yields an overall wall-to-loadefficiency of ˜15%, which is reasonable given the expected ˜40%efficiency for the wireless power transmission at that distance and thelow efficiency of the driving circuit.

From the theoretical treatment above, we see that in typical examples itis important that the coils be on resonance for the power transmissionto be practical. We found experimentally that the power transmitted tothe load dropped sharply as one of the coils was detuned from resonance.For a fractional detuning Δf/f₀ of a few times the inverse loaded Q, theinduced current in the device coil was indistinguishable from noise.

The power transmission was not found to be visibly affected as humansand various everyday objects, such as metallic and wooden furniture, aswell as electronic devices large and small, were placed between the twocoils, even when they drastically obstructed the line of sight betweensource and device. External objects were found to have an effect onlywhen they were closer than 10 cm from either one of the coils. Whilesome materials (such as aluminum foil, styrofoam and humans) mostly justshifted the resonant frequency, which could in principle be easilycorrected with a feedback circuit of the type described earlier, others(cardboard, wood, and PVC) lowered Q when placed closer than a fewcentimeters from the coil, thereby lowering the efficiency of thetransfer.

This method of power transmission is believed safe for humans. Whentransmitting 60 W (more than enough to power a laptop computer) across 2m, we estimated that the magnitude of the magnetic field generated ismuch weaker than the Earth's magnetic field for all distances except forless than about 1 cm away from the wires in the coil, an indication ofthe safety of the scheme even after long-term use. The power radiatedfor these parameters was ˜5 W, which is roughly an order of magnitudehigher than cell phones but could be drastically reduced, as discussedbelow.

Although the two coils are currently of identical dimensions, it ispossible to make the device coil small enough to fit into portabledevices without decreasing the efficiency. One could, for instance,maintain the product of the characteristic sizes of the source anddevice coils constant.

These experiments demonstrated experimentally a system for powertransmission over medium range distances, and found that theexperimental results match theory well in multiple independent andmutually consistent tests.

The efficiency of the scheme and the distances covered can be improvedby silver-plating the coils, which may increase their Q, or by workingwith more elaborate geometries for the resonant objects. Nevertheless,the performance characteristics of the system presented here are alreadyat levels where they could be useful in practical applications.

2.2 Capacitively-Loaded Conducting Loops or Coils

In some examples, one or more of the resonant objects arecapacitively-loaded conducting loops or coils. Referring to FIG. 9 ahelical coil with N turns of conducting wire, as described above, isconnected to a pair of conducting parallel plates of area A spaced bydistance d via a dielectric material of relative permittivity ∈, andeverything is surrounded by air (as shown, N=1 and h=0). The plates havea capacitance C_(p)=∈_(o)∈A/d, which is added to the distributedcapacitance of the coil and thus modifies its resonance. Note however,that the presence of the loading capacitor may modify the currentdistribution inside the wire and therefore the total effectiveinductance L and total effective capacitance C of the coil may bedifferent respectively from L_(s) and C_(s), which are calculated for aself-resonant coil of the same geometry using a sinusoidal currentprofile. Since some charge may accumulate at the plates of the externalloading capacitor, the charge distribution ρ inside the wire may bereduced, so C<C_(s), and thus, from the charge conservation equation,the current distribution j may flatten out, so L>L_(s). The resonantfrequency for this system may be ω=1/√{square root over(L(C+C_(p)))}<ω_(s)=1/√{square root over (L_(s)C_(s))}, and I(x)→I_(o)cos(πx/l)

C→C_(s)

ω→ω_(s), as C_(p)→0.

In general, the desired CMT parameters can be found for this system, butagain a very complicated solution of Maxwell's Equations is required.Instead, we will analyze only a special case, where a reasonable guessfor the current distribution can be made. When C_(p)

C_(s)>C, then ω≈1/√{square root over (LC_(p))}

ω_(s) and Z≈√{square root over (L/C_(p))}

Z_(s), while all the charge is on the plates of the loading capacitorand thus the current distribution is constant along the wire. Thisallows us now to compute numerically L from Eq. (19). In the case h=0and N integer, the integral in Eq. (19) can actually be computedanalytically, giving the formula L=μ_(o)r[ln(8r/a)−2]N². Explicitanalytical formulas are again available for R from Eq. (21) and (22),since I_(rms)=I_(o), |p|≈0 and |m|=I_(o)Nπr² (namely only themagnetic-dipole term is contributing to radiation), so we can determinealso Q_(abs)=ωL/R_(abs) and Q_(rad)=ωL/R_(rad). At the end of thecalculations, the validity of the assumption of constant current profileis confirmed by checking that indeed the condition C_(p)

C_(s)

ω

ω_(s) is satisfied. To satisfy this condition, one could use a largeexternal capacitance, however, this would usually shift the operationalfrequency lower than the optimal frequency, which we will determineshortly; instead, in typical examples, one often prefers coils with verysmall self-capacitance C_(s) to begin with, which usually holds, for thetypes of coils under consideration, when N=1, so that theself-capacitance comes from the charge distribution across the singleturn, which is almost always very small, or when N>1 and h

2Na, so that the dominant self-capacitance comes from the chargedistribution across adjacent turns, which is small if the separationbetween adjacent turns is large.

The external loading capacitance C_(p) provides the freedom to tune theresonant frequency (for example by tuning A or d). Then, for theparticular simple case h=0, for which we have analytical formulas, thetotal Q=ωL/(R_(abs)+R_(rad)) becomes highest at the optimal frequency

$\begin{matrix}{{\omega_{Q\;} = \left\lbrack {\frac{c^{4}}{\pi}{\sqrt{\frac{ɛ_{o}}{2\sigma}} \cdot \frac{1}{a\; N\; r^{3}}}} \right\rbrack^{2/7}},} & (24)\end{matrix}$reaching the value

$\begin{matrix}{Q_{\max} = {\frac{6}{7\pi}{\left( {2\pi^{2}\eta_{o}\frac{\sigma\; a^{2}N^{2}}{r}} \right)^{3/7} \cdot {\left\lbrack {{\ln\left( \frac{8r}{a} \right)} - 2} \right\rbrack.}}}} & (25)\end{matrix}$

At lower frequencies Q is dominated by ohmic loss and at higherfrequencies by radiation. Note, however, that the formulas above areaccurate as long as ω_(Q)

ω_(s) and, as explained above, this holds almost always when N=1, and isusually less accurate when N>1, since h=0 usually implies a largeself-capacitance. A coil with large h can be used, if theself-capacitance needs to be reduced compared to the externalcapacitance, but then the formulas for L and ω_(Q), Q_(max) are againless accurate. Similar qualitative behavior is expected, but a morecomplicated theoretical model is needed for making quantitativepredictions in that case.

The results of the above analysis for two examples of subwavelengthmodes of λ/r≧70 (namely highly suitable for near-field coupling and wellwithin the quasi-static limit) of coils with N=1 and h=0 at the optimalfrequency Eq. (24) are presented in Table 3. To confirm the validity ofconstant-current assumption and the resulting analytical formulas,mode-solving calculations were also performed using another completelyindependent method: computational 3D finite-element frequency-domain(FEFD) simulations (which solve Maxwell's Equations in frequency domainexactly apart for spatial discretization) were conducted, in which theboundaries of the conductor were modeled using a complex impedanceζ_(c)=√{square root over (μ_(o)ω/2σ)} boundary condition, valid as longas ζ_(c)/ζ_(o)

1 (<10⁻⁵ for copper in the microwave). Table 3 shows Numerical FEFD (andin parentheses analytical) results for the wavelength and absorption,radiation and total loss rates, for two different cases ofsubwavelength-loop resonant modes. Note that copper was used for theconducting material (6=5.998·10⁷ S/m). Specific parameters of the plotin FIG. 4 are highlighted in bold in the table. The two methods(analytical and computational) are in good agreement and show that, insome examples, the optimal frequency is in the low-MHz microwave rangeand the expected quality factors are Q_(abs)≧1000 and Q_(rad)≧10000.

TABLE 3 single coil λ/r f Q_(rad) Q_(abs) Q r = 30 cm, a = 2 cm 111.4(112.4) 8.976 (8.897) 29546 (30512) 4886 (5117) 4193 (4381) ε = 10, A =138 cm², d = 4 mm r = 10 cm, a = 2 mm  69.7 (70.4) 43.04 (42.61) 10702(10727) 1545 (1604) 1350 (1395) ε = 10, A = 3.14 cm², d = 1 mm

Referring to FIG. 10, in some examples, energy is transferred betweentwo capacitively-loaded coils. For the rate of energy transfer betweentwo capacitively-loaded coils 1 and 2 at distance D between theircenters, the mutual inductance M_(L) can be evaluated numerically fromEq. (23) by using constant current distributions in the case ω

ω_(s). In the case h=0, the coupling may be only magnetic and again wehave an analytical formula, which, in the quasi-static limit r<<D<<λ andfor the relative orientation shown in FIG. 10, isM_(L)≈πμ_(o)/2·(r₁r₂)²N₁N₂/D³, which means that k∝(√{square root over(r₁r₂)}/D)³ may be independent of the frequency ω and the number ofturns N₁, N₂. Consequently, the resultant coupling figure-of-merit ofinterest is

$\begin{matrix}{{U = {{k\sqrt{Q_{1}Q_{2}}} \approx {\left( \frac{\sqrt{r_{1}r_{2}}}{D} \right)^{3} \cdot \frac{\pi^{2}\eta_{o}{\frac{\sqrt{r_{1}r_{2}}}{\lambda} \cdot N_{1}}N_{2}}{\prod\limits_{{j = 1},2}\;\left( {{{\sqrt{\frac{{\pi\eta}_{o}}{\lambda\sigma}} \cdot \frac{r_{j}}{a_{j}}}N_{j}} + {\frac{8}{3}\pi^{5}{\eta_{o}\left( \frac{r_{j}}{\lambda} \right)}^{4}N_{j}^{2}}} \right)^{1/2}}}}},} & (26)\end{matrix}$which again is more accurate for N₁=N₂=1.

From Eq. (26) it can be seen that the optimal frequency ω_(U), where thefigure-of-merit is maximized to the value U_(max), is close to thefrequency ω_(Q) ₁ _(Q) ₂ at which Q₁Q₂ is maximized, since k does notdepend much on frequency (at least for the distances D<<λ of interestfor which the quasi-static approximation is still valid). Therefore, theoptimal frequency ω_(U)≈ω_(Q) ₁ _(Q) ₂ may be mostly independent of thedistance D between the two coils and may lie between the two frequenciesω_(Q) ₁ at and ω_(Q) ₂ at which the single-coil Q₁ and Q₂ respectivelypeak. For same coils, this optimal frequency is given by Eq. (24) andthen the strong-coupling factor from Eq. (26) becomes

$\begin{matrix}{U_{\max} = {{kQ}_{\max} \approx {{\left( \frac{r}{D} \right)^{3} \cdot \frac{3}{7}}{\left( {2\pi^{2}\eta_{o}\frac{\sigma\; a^{2}N^{2}}{r}} \right)^{3/7}.}}}} & (27)\end{matrix}$

In some examples, one can tune the capacitively-loaded conducting loopsor coils, so that their angular resonant frequencies are close to ω_(U)within Γ_(U), which is half the angular frequency width for whichU>U_(max)/2.

Referring to Table 4, numerical FEFD and, in parentheses, analyticalresults based on the above are shown for two systems each composed of amatched pair of the loaded coils described in Table 3. The averagewavelength and loss rates are shown along with the coupling rate andcoupling to loss ratio figure-of-merit U=κ/Γ as a function of thecoupling distance D, for the two cases. Note that the average numericalΓ_(rad) shown are slightly different from the single-loop value of FIG.3. Analytical results for Γ_(rad) are not shown but the single-loopvalue is used. (The specific parameters corresponding to the plot inFIG. 10 are highlighted with bold in the table.) Again we chose N=1 tomake the constant-current assumption a good one and computed M_(L)numerically from Eq. (23). Indeed the accuracy can be confirmed by theiragreement with the computational FEFD mode-solver simulations, whichgive κ through the frequency splitting of the two normal modes of thecombined system (δ_(E)=2κ from Eq. (4)). The results show that formedium distances D/r=10−3 the expected coupling-to-loss ratios are inthe range U˜0.5−50.

TABLE 4 pair of coils D/r Q^(rad) Q = ω/2Γ Q_(κ) = ω/2κ κ/Γ r = 30 cm, a= 2 cm 3 30729 4216 62.6 (63.7) 67.4 (68.7) ε = 10, A = 138 cm², 5 295774194 235 (248) 17.8 (17.6) d = 4 mm 7 29128 4185 589 (646) 7.1 (6.8) λ/r≈ 112 10 28833 4177 1538 (1828) 2.7 (2.4) Q^(abs) ≈ 4886 r = 10 cm, a =2 mm 3 10955 1355 85.4 (91.3) 15.9 (15.3) ε = 10, A = 5 10740 1351 313(356) 4.32 (3.92) 3.14 cm², d = 1 mm 7 10759 1351 754 (925) 1.79 (1.51)λ/r ≈ 70 10 10756 1351 1895 (2617) 0.71 (0.53) Q^(abs) ≈  15462.2.1 Derivation of Optimal Power-Transmission Efficiency

-   -   Referring to FIG. 11, to rederive and express Eq. (15) in terms        of the parameters which are more directly accessible from        particular resonant objects, such as the capacitively-loaded        conducting loops, one can consider the following circuit-model        of the system, where the inductances L_(s),L_(d) represent the        source and device loops respectively, R_(s),R_(d) their        respective losses, and C_(s), C_(d) are the required        corresponding capacitances to achieve for both resonance at        frequency ω. A voltage generator V_(g) is considered to be        connected to the source and a load resistance R_(l) to the        device. The mutual inductance is denoted by M.

Then from the source circuit at resonance (ωL_(s)=1/ωC_(s)):

$\begin{matrix}{{V_{g} = {\left. {{I_{s}R_{s}} - {{j\omega}\;{MI}_{d}}}\Rightarrow{\frac{1}{2}V_{g}^{*}I_{s}} \right. = {{\frac{1}{2}{I_{s}}^{2}R_{s}} + {\frac{1}{2}{j\omega}\;{MI}_{d}^{*}I_{s}}}}},} & (28)\end{matrix}$and from the device circuit at resonance (ωL_(d)=1/ωC_(d)):0=I _(d)(R _(d) +R _(l))−jωMI _(s)

jωMI_(s) =I _(d)(R _(d) +R _(l))  (29)

So by substituting Eq. (29) to Eq. (28) and taking the real part (fortime-averaged power) we get:

$\begin{matrix}{{P_{g} = {{{Re}\left\{ {\frac{1}{2}V_{g}^{*}I_{s}} \right\}} = {{{\frac{1}{2}{I_{s}}^{2}R_{s}} + {\frac{1}{2}{I_{d}}^{2}\left( {R_{d} + R_{l}} \right)}} = {P_{s} + P_{d} + P_{l}}}}},} & (30)\end{matrix}$where we identified the power delivered by the generatorP_(g)=Re{V_(g)*I_(s)/2}, the power lost inside the sourceP_(s)=|I_(s)|²R_(s)/2, the power lost inside the deviceP_(d)=|I_(d)|²R_(d)/2 and the power delivered to the loadP_(l)=|I_(d)|²R_(l)/2. Then, the power transmission efficiency is:

$\begin{matrix}{{\eta_{P} \equiv \frac{P_{l}}{P_{g}}} = {\frac{R_{l}}{{{\frac{I_{s}}{I_{d}}}^{2}R_{s}} + \left( {R_{d} + R_{l}} \right)}\overset{(29)}{=}{\frac{R_{l}}{{\frac{\left( {R_{d} + R_{l}} \right)^{2}}{\left( {\omega\; M} \right)^{2}}R_{s}} + \left( {R_{d} + R_{l}} \right)}.}}} & (31)\end{matrix}$

If we now choose the load impedance R_(l) to optimize the efficiency byη_(P)′(R_(l))=0, we get the optimal load impedance

$\begin{matrix}{\frac{R_{l*}}{R_{d}} = \sqrt{1 + \frac{\left( {\omega\; M} \right)^{2}}{R_{s}R_{d}}}} & (32)\end{matrix}$and the maximum possible efficiency

$\begin{matrix}{\eta_{P*} = {\frac{{R_{l*}/R_{d}} - 1}{{R_{l*}/R_{d}} + 1} = {\left\lbrack \frac{\omega\; M\text{/}\sqrt{R_{s}R_{d}}}{1 + \sqrt{1 + \left( {\omega\; M\text{/}\sqrt{R_{s}R_{d}}} \right)^{2}}} \right\rbrack^{2}.}}} & (33)\end{matrix}$

To check now the correspondence with the CMT model, note thatκ_(l)=R_(l)/2L_(d), Γ_(d)=R_(d)/2L_(d), Γ_(s)=R_(S)/2L_(s), andκ=ωM/2√{square root over (L_(s)L_(d))}, so thenU_(l)=κ_(l)/δ_(d)=R_(l)/R_(d) and U=κ/√{square root over(Γ_(s)Γ_(d))}=ωM/√{square root over (R_(s)R_(d))}. Therefore, thecondition Eq. (32) is identical to the condition Eq. (14) and theoptimal efficiency Eq. (33) is identical to the general Eq. (15).Indeed, as the CMT analysis predicted, to get a large efficiency, weneed to design a system that has a large strong-coupling factor U.

2.2.2 Optimization of U

The results above can be used to increase or optimize the performance ofa wireless energy transfer system, which employs capacitively-loadedcoils. For example, from the scaling of Eq. (27) with the differentsystem parameters, one sees that to maximize the system figure-of-meritU, in some examples, one can:

-   -   Decrease the resistivity of the conducting material. This can be        achieved, for example, by using good conductors (such as copper        or silver) and/or lowering the temperature. At very low        temperatures one could use also superconducting materials to        achieve extremely good performance.    -   Increase the wire radius a. In typical examples, this action can        be limited by physical size considerations. The purpose of this        action is mainly to reduce the resistive losses in the wire by        increasing the cross-sectional area through which the electric        current is flowing, so one could alternatively use also a Litz        wire, or ribbon, or any low AC-resistance structure, instead of        a circular wire.    -   For fixed desired distance D of energy transfer, increase the        radius of the loop r. In typical examples, this action can be        limited by physical size considerations.    -   For fixed desired distance vs. loop-size ratio D/r, decrease the        radius of the loop r. In typical examples, this action can be        limited by physical size considerations.    -   Increase the number of turns N. (Even though Eq. (27) is        expected to be less accurate for N>1, qualitatively it still        provides a good indication that we expect an improvement in the        coupling-to-loss ratio with increased N.) In typical examples,        this action can be limited by physical size and possible voltage        considerations, as will be discussed in following paragraphs.    -   Adjust the alignment and orientation between the two coils. The        figure-of-merit is optimized when both cylindrical coils have        exactly the same axis of cylindrical symmetry (namely they are        “facing” each other). In some examples, particular mutual coil        angles and orientations that lead to zero mutual inductance        (such as the orientation where the axes of the two coils are        perpendicular and the centers of the two coils are on one of the        two axes) should be avoided.    -   Finally, note that the height of the coil h is another available        design parameter, which can have an impact on the performance        similar to that of its radius r, and thus the design rules can        be similar.

The above analysis technique can be used to design systems with desiredparameters. For example, as listed below, the above described techniquescan be used to determine the cross sectional radius a of the wire usedto design a system including two same single-turn loops with a givenradius in order to achieve a specific performance in terms of U=κ/Γ at agiven D/r between them, when the loop material is copper (σ=5.998·10⁷S/m):

-   -   D/r=5, U≧10, r=30 cm        a≧9 mm    -   D/r=5, U≧10, r=5 cm        a≧3.7 mm    -   D/r=5, U≧20, r=30 cm        a≧20 mm    -   D/r=5, U≧20, r=5 cm        a≧8.3 mm    -   D/r=10, U≧1, r=30 cm        a≧7 mm    -   D/r=10, U≧1, r=5 cm        a≧2.8 mm    -   D/r=10, U≧3, r=30 cm        a≧25 mm    -   D/r=10, U≧3, r=5 cm        a≧10 mm

Similar analysis can be done for the case of two dissimilar loops. Forexample, in some examples, the device under consideration may beidentified specifically (e.g. a laptop or a cell phone), so thedimensions of the device object (r_(d),h_(d),a_(d),N_(d)) may berestricted. However, in some such examples, the restrictions on thesource object (r_(s),h_(s),a_(s),N_(s)) may be much less, since thesource can, for example, be placed under the floor or on the ceiling. Insuch cases, the desired distance between the source and device may befixed; in other cases it may be variable. Listed below are examples(simplified to the case N_(s)=N_(d)=1 and h_(s)=h_(d)=0) of how one canvary the dimensions of the source object to achieve a desired systemperformance in terms of U_(sd)=κ/√{square root over (Γ_(s)Γ_(d))}, whenthe material is again copper (σ=5.998·10⁷ S/m):

-   -   D=1.5 m, U_(sd)≧15, r_(d)=30 cm, a_(d)=6 mm        r_(s)=1.158 m, a_(s)≧5 mm    -   D=1.5 m, U_(sd)≧30, r_(d)=30 cm, a_(d)=6 mm        r_(s)=1.15 m, a_(s)≧33 mm    -   D=1.5 m, U_(sd)≧1, r_(d)=5 cm, a_(d)=4 mm        r_(s)=1.119 m, a_(s)≧7 mm    -   D=1.5 m, U_(sd)≧2, r_(d)=5 cm, a_(d)=4 mm        r_(s)=1.119 m, a, ≧52 mm    -   D=2 m, U_(sd)≧10, r_(d)=30 cm, a_(d)=6 mm        r_(s)=1.518 m, a_(s)≧7 mm    -   D=2 m, U_(sd)≧20, r_(d)=30 cm, a_(d)=6 mm        r_(s)=1.514 m, a_(s)≧50 mm    -   D=2 m, U_(sd)≧0.5, r_(d)=5 cm, a_(d)=4 mm        r_(s)=1.491 m, a_(s)≧5 mm    -   D=2 m, U_(sd)≧1, r_(d)=5 cm, a_(d)=4 mm        r_(s)=1.491 m, a_(s)≧36 mm        2.2.3 Optimization of k

As described below, in some examples, the quality factor Q of theresonant objects may be limited from external perturbations and thusvarying the coil parameters may not lead to significant improvements inQ. In such cases, one can opt to increase the strong-coupling factor Uby increasing the coupling factor k. The coupling does not depend on thefrequency and may weakly depend on the number of turns. Therefore, insome examples, one can:

-   -   Increase the wire radii a₁ and a₂. In typical examples, this        action can be limited by physical size considerations.    -   For fixed desired distance D of energy transfer, increase the        radii of the coils r₁ and r₂. In typical examples, this action        can be limited by physical size considerations.    -   For fixed desired distance vs. coil-sizes ratio D/√{square root        over (r₁r₂)}, only the weak (logarithmic) dependence of the        inductance remains, which suggests that one should decrease the        radii of the coils r₁ and r₂. In typical examples, this action        can be limited by physical size considerations.    -   Adjust the alignment and orientation between the two coils. In        typical examples, the coupling is optimized when both        cylindrical coils have exactly the same axis of cylindrical        symmetry (namely they are “facing” each other). Particular        mutual coil angles and orientations that lead to zero mutual        inductance (such as the orientation where the axes of the two        coils are perpendicular and the centers of the two coils are on        one of the two axes) should obviously be avoided.    -   Finally, note that the heights of the coils h₁ and h₂ are other        available design parameters, which can have an impact to the        coupling similar to that of their radii r₁ and r₂, and thus the        design rules can be similar.

Further practical considerations apart from efficiency, e.g. physicalsize limitations, will be discussed in detail below.

2.2.4 Optimization of Overall System Performance

In embodiments, the dimensions of the resonant objects may be determinedby the particular application. For example, when the application ispowering a laptop or a cell-phone, the device resonant object cannothave dimensions that exceed those of the laptop or cell-phonerespectively. For a system of two loops of specified dimensions, interms of loop radii r_(s,d) and wire radii a_(s,d), the independentparameters left to adjust for the system optimization are: the number ofturns N_(s,d), the frequency f, the power-load consumption rateκ_(l)=R_(l)/2L_(d) and the power-generator feeding rateκ_(g)=R_(g)/2L_(s), where R_(g) is the internal (characteristic)impedance of the generator.

In general, in various examples, the dependent variable that one maywant to increase or optimize may be the overall efficiency η. However,other important variables may need to be taken into consideration uponsystem design. For example, in examples featuring capacitively-loadedcoils, the designs can be constrained by, the currents flowing insidethe wires I_(s,d) and other components and the voltages across thecapacitors V_(s,d). These limitations can be important because for ˜Wattpower applications the values for these parameters can be too large forthe wires or the capacitors respectively to handle. Furthermore, thetotal loaded (by the load) quality factor of the deviceQ_(d[l])=ω/2(Γ_(d)+Γ_(l))=ωL_(d)/(R_(d)+R_(l)) and the total loaded (bythe generator) quality factor of the sourceQ_(s[g])=ω/2(Γ_(s)+Γ_(g))=ωL_(s)/(R_(s)+R_(g)) are quantities thatshould be preferably small, because to match the source and deviceresonant frequencies to within their Q's, when those are very large, canbe challenging experimentally and more sensitive to slight variations.Lastly, the radiated powers P_(s,rad) and P_(d,rad) may need to beminimized for concerns about far-field interference and safety, eventhough, in general, for a magnetic, non-radiative scheme they arealready typically small. In the following, we examine then the effectsof each one of the independent variables on the dependent ones.

We define a new variable wp to express the power-load consumption ratefor some particular value of U through U_(l)=κ_(l)/Γ_(d)=√{square rootover (1+wp·U²)}. Then, in some examples, values which may impact thechoice of this rate are: U_(l)=1

wp=0 to minimize the required energy stored in the source (and thereforeI_(s) and V_(s)), U_(l)=√{square root over (1+U²)}>1

wp=1 to maximize the efficiency, as seen earlier, or U_(l)

1

wp

1 to decrease the required energy stored in the device (and thereforeI_(d) and V_(d)) and to decrease or minimize Q_(d[l]). Similar is theimpact of the choice of the power-generator feeding rateU_(g)=κ_(g)/Γ_(s), with the roles of the source/device andgenerator/load reversed.

In some examples, increasing N_(s) and N_(d) may increase Q_(s) andQ_(d), and thus U and the efficiency significantly. It also may decreasethe currents I_(s) and I_(d), because the inductance of the loops mayincrease, and thus the energy W_(s,d)=L_(s,d)|I_(s,d)|²/2 required forgiven output power P_(l) can be achieved with smaller currents. However,in some examples, increasing N_(d) and thus Q_(d) can increase Q_(d[l]),P_(d,rad) and the voltage across the device capacitance V_(d). Similarcan be the impact of increasing N_(s) on Q_(s[g]), P_(s,rad) and V_(s).As a conclusion, in some examples, the number of turns N_(s) and N_(d)may be chosen large enough (for high efficiency) but such that theyallow for reasonable voltages, loaded Q's and/or powers radiated.

With respect to the resonant frequency, again, there may be an optimalone for efficiency. Close to that optimal frequency Q_(d[l]) and/orQ_(s[g]) can be approximately maximum. In some examples, for lowerfrequencies the currents may typically get larger but the voltages andradiated powers may get smaller. In some examples, one may choose theresonant frequency to maximize any of a number of system parameters orperformance specifications, such as efficiency.

One way to decide on an operating regime for the system may be based ona graphical method. Consider two loops of r_(s)=25 cm, r_(d)=15 cm,h_(s)=h_(d)=0, a_(s)=a_(d)=3 mm and distance D=2 m between them. In FIG.12, we plot some of the above dependent variables (currents, voltagesand radiated powers normalized to 1 Watt of output power) in terms offrequency f and N_(d), given some choice for wp and N_(s). FIG. 12depicts the trend of system performance explained above. In FIG. 13, wemake a contour plot of the dependent variables as functions of bothfrequency and wp but for both N_(s) and N_(d) fixed. For example, inembodiments, a reasonable choice of parameters for the system of twoloops with the dimensions given above may be: N_(s)=2, N_(d)=6, f=10 MHzand wp=10, which gives the following performance characteristics:η=20.6%, Q_(d[l])=1264, I_(s)=7.2 A, I_(d)=1.4 A, V_(s)=2.55 kV,V_(d)=2.30 kV, P_(s,rad)=0.15 W, P_(d,rad)=0.006 W. Note that theresults in FIGS. 12, 13 and the calculated performance characteristicsare made using the analytical formulas provided above, so they areexpected to be less accurate for large values of N_(s), N_(d), but stillmay give a good estimate of the scalings and the orders of magnitude.

Finally, in embodiments, one could additionally optimize for the sourcedimensions, if, for example, only the device dimensions are limited.Namely, one can add r_(s) and a_(s) in the set of independent variablesand optimize with respect to these all the dependent variables of thesystem. In embodiments, such an optimization may lead to improvedresults.

2.3 Inductively-Loaded Conducting Rods

A straight conducting rod of length 2h and cross-sectional radius a hasdistributed capacitance and distributed inductance, and therefore cansupport a resonant mode of angular frequency ω. Using the same procedureas in the case of self-resonant coils, one can define an effective totalinductance L and an effective total capacitance C of the rod throughformulas Eqs. (19) and (20). With these definitions, the resonantangular frequency and the effective impedance may be given again by thecommon formulas ω=1/√{square root over (LC)} and Z=√{square root over(L/C)} respectively. To calculate the total inductance and capacitance,one can assume again a sinusoidal current profile along the length ofthe conducting wire. When interested in the lowest mode, if we denote byx the coordinate along the conductor, such that it runs from −h to +h,then the current amplitude profile may have the form I(x)=I_(o)cos(πx/2h), since it has to be zero at the open ends of the rod. This isthe well-known half-wavelength electric dipole resonant mode.

In some examples, one or more of the resonant objects may beinductively-loaded conducting rods. Referring to FIG. 14, a straightconducting rod of length 2h and cross-sectional radius a, as in theprevious paragraph, is cut into two equal pieces of length h, which maybe connected via a coil wrapped around a magnetic material of relativepermeability μ, and everything is surrounded by air. The coil has aninductance L_(c), which is added to the distributed inductance of therod and thus modifies its resonance. Note however, that the presence ofthe center-loading inductor may modify the current distribution insidethe wire and therefore the total effective inductance L and totaleffective capacitance C of the rod may be different respectively fromL_(s) and C_(s), which are calculated for a self-resonant rod of thesame total length using a sinusoidal current profile, as in the previousparagraph. Since some current may be running inside the coil of theexternal loading inductor, the current distribution j inside the rod maybe reduced, so L<L_(s), and thus, from the charge conservation equation,the linear charge distribution ρ_(l) may flatten out towards the center(being positive in one side of the rod and negative in the other side ofthe rod, changing abruptly through the inductor), so C>C_(s). Theresonant frequency for this system may be ω=1/√{square root over((L+L_(c))C)}<ω_(s)=1/√{square root over (L_(s)C_(s))}, and I(x)→I_(o)cos(πx/2h)

L→L_(s)

ω→ω_(s), as L_(c)→0.

In general, the desired CMT parameters can be found for this system, butagain a very complicated solution of Maxwell's Equations is generallyrequired. In a special case, a reasonable estimate for the currentdistribution can be made. When L_(c)

L_(s)>L, then ω≈1/√{square root over (L_(c)C)}

ω_(s) and Z≈√{square root over (L_(c)/C)}

Z_(s), while the current distribution is triangular along the rod (withmaximum at the center-loading inductor and zero at the ends) and thusthe charge distribution may be positive constant on one half of the rodand equally negative constant on the other side of the rod. This allowsus to compute numerically C from Eq. (20). In this case, the integral inEq. (20) can actually be computed analytically, giving the formula1/C=1/(π∈_(o)h)[ln(h/a)−1]. Explicit analytical formulas are againavailable for R from Eq. (21) and (22), since I_(rms)=I_(o), |p|=q_(o)hand |m|=0 (namely only the electric-dipole term is contributing toradiation), so we can determine also Q_(abs)=1/ωCR_(abs) andQ_(rad)=1/ωCR_(rad). At the end of the calculations, the validity of theassumption of triangular current profile may be confirmed by checkingthat indeed the condition L_(c)

L_(s)

ω

ω_(s) is satisfied. This condition may be relatively easily satisfied,since typically a conducting rod has very small self-inductance L_(s) tobegin with.

Another important loss factor in this case is the resistive loss insidethe coil of the external loading inductor L_(c) and it may depend on theparticular design of the inductor. In some examples, the inductor may bemade of a Brooks coil, which is the coil geometry which, for fixed wirelength, may demonstrate the highest inductance and thus quality factor.The Brooks coil geometry has N_(Bc) turns of conducting wire ofcross-sectional radius a_(Bc) wrapped around a cylindrically symmetriccoil former, which forms a coil with a square cross-section of sider_(Bc), where the inner side of the square is also at radius) r_(Bc)(and thus the outer side of the square is at radius 2r_(Bc)), thereforeN_(Bc)≈(r_(Bc)/2a_(Bc))². The inductance of the coil is thenL_(c)=2.0285μ_(o)r_(Bc)N_(Bc) ²≈2.0285μ_(o)r_(Bc) ⁵/8a_(Bc) ⁴ and itsresistance

${R_{c} \approx {\frac{1}{\sigma}\frac{l_{Bc}}{\pi\; a_{Bc}^{2}}\sqrt{1 + {\frac{\mu_{o}\omega\;\sigma}{2}\left( \frac{a_{Bc}}{2} \right)^{2}}}}},$where the total wire length is l_(Bc)≈2π(3r_(Bc)/2)N_(Bc)≈3πr_(Bc)³/4a_(Bc) ² and we have used an approximate square-root law for thetransition of the resistance from the dc to the ac limit as the skindepth varies with frequency.

The external loading inductance L_(c) provides the freedom to tune theresonant frequency. For example, for a Brooks coil with a fixed sizer_(Bc), the resonant frequency can be reduced by increasing the numberof turns N_(Bc) by decreasing the wire cross-sectional radius a_(Bc).Then the desired resonant angular frequency ω=1/√{square root over(L_(c)C)} may be achieved for a_(Bc)≈(2.0285μ_(o)r_(Bc) ⁵ω²C)^(1/4) andthe resulting coil quality factor may be

$Q_{c} \approx {0.169\;\mu_{o}\sigma\; r_{Bc}^{2}{\omega/{\sqrt{1 + {\omega^{2}\mu_{o}\sigma\sqrt{2.0285\mspace{14mu}{\mu_{0}\left( {r_{Bc}/4} \right)}^{5}C}}}.}}}$Then, for the particular simple case L_(c)

L_(s), for which we have analytical formulas, the totalQ=1/ωC(R_(c)+R_(abs)+R_(rad)) becomes highest at some optimal frequencyω_(Q), reaching the value Q_(max), that may be determined by theloading-inductor specific design. For example, for the Brooks-coilprocedure described above, at the optimal frequencyQ_(max)≈Q_(c)≈0.8(μ_(o)σ²r_(Bc) ³/C)^(1/4). At lower frequencies it isdominated by ohmic loss inside the inductor coil and at higherfrequencies by radiation. Note, again, that the above formulas areaccurate as long as ω_(Q)

ω_(s) and, as explained above, this may be easy to design for by using alarge inductance.

The results of the above analysis for two examples, using Brooks coils,of subwavelength modes of λ/h≧200 (namely highly suitable for near-fieldcoupling and well within the quasi-static limit) at the optimalfrequency ω_(Q) are presented in Table 5.

Table 5 shows in parentheses (for similarity to previous tables)analytical results for the wavelength and absorption, radiation andtotal loss rates, for two different cases of subwavelength-rod resonantmodes. Note that copper was used for the conducting material(σ=5.998·10⁷ S/m). The results show that, in some examples, the optimalfrequency may be in the low-MHz microwave range and the expected qualityfactors may be Q_(abs)≧1000 and Q_(rad)≧100000.

TABLE 5 single rod λ/h f (MHz) Q_(rad) Q_(abs) Q h = 30 cm, a = 2 cm(403.8)  (2.477) (2.72 * 10⁶) (7400) (7380) μ = 1, r_(Bc) = 2 cm, a_(Bc)= 0.88 mm, N_(Bc) = 129 h = 10 cm, a = 2 mm (214.2) (14.010) (6.92 *10⁵) (3908) (3886) μ = 1, r_(Bc) = 5 mm, a_(Bc) = 0.25 mm, N_(Bc) = 103

In some examples, energy may be transferred between twoinductively-loaded rods. For the rate of energy transfer between twoinductively-loaded rods 1 and 2 at distance D between their centers, themutual capacitance M_(C) can be evaluated numerically from Eq. (23) byusing triangular current distributions in the case ω

ω_(s). In this case, the coupling may be only electric and again we havean analytical formula, which, in the quasi-static limit h<<D<<λ and forthe relative orientation such that the two rods are aligned on the sameaxis, is 1/M_(C)≈½π∈_(o)·(h₁h₂)²/D³, which means that k∝(√{square rootover (h₁h₂)}/D)³ is independent of the frequency ω. One can then get theresultant strong-coupling factor U.

It can be seen that the frequency ω_(U), where the figure-of-merit ismaximized to the value U_(max), is close to the frequency ω_(Q) ₁ _(Q) ₂, where Q₁Q₂ is maximized, since k does not depend much on frequency (atleast for the distances D<<λ of interest for which the quasi-staticapproximation is still valid). Therefore, the optimal frequency ω_(U)≈ωQ₁ _(Q) ₂ may be mostly independent of the distance D between the tworods and may lie between the two frequencies ω_(Q) ₁ and ω_(Q) ₂ atwhich the single-rod Q₁ and Q₂ respectively peak. In some typicalexamples, one can tune the inductively-loaded conducting rods, so thattheir angular eigenfrequencies are close to ω_(U) within Γ_(U), which ishalf the angular frequency width for which U>U_(max)/2.

Referring to Table 6, in parentheses (for similarity to previous tables)analytical results based on the above are shown for two systems eachcomposed of a matched pair of the loaded rods described in Table 5. Theaverage wavelength and loss rates are shown along with the coupling rateand coupling to loss ratio figure-of-merit U=κ/Γ as a function of thecoupling distance D, for the two cases. Note that for Γ_(rad) thesingle-rod value is used. Again we chose L_(c)

L_(s) to make the triangular-current assumption a good one and computedM_(C) numerically from Eq. (23). The results show that for mediumdistances D/h=10−3 the expected coupling-to-loss ratios are in the rangeU˜0.5-100.

TABLE 6 pair of rods D/h Q_(κ) = 1/k U h = 30 cm, a = 2 cm 3  (70.3)(105.0)  μ = 1, r_(Bc) = 2 cm, a_(Bc) = 0.88 mm, N_(Bc) = 129 5  (389)(19.0) λ/h ≈ 404 7 (1115) (6.62) Q ≈ 7380 10 (3321) (2.22) h = 10 cm, a= 2 mm 3  (120) (32.4) μ = 1, r_(Bc) = 5 mm, a_(Bc) = 0.25 mm, N_(Bc) =103 5  (664) (5.85) λ/h ≈ 214 7 (1900) (2.05) Q ≈ 3886 10 (5656) (0.69)2.4 Dielectric Disks

-   -   In some examples, one or more of the resonant objects are        dielectric objects, such as disks. Consider a two dimensional        dielectric disk object, as shown in FIG. 15( a), of radius r and        relative permittivity ∈ surrounded by air that supports high-Q        “whispering-gallery” resonant modes. The loss mechanisms for the        energy stored inside such a resonant system are radiation into        free space and absorption inside the disk material. High-Q_(rad)        and long-tailed subwavelength resonances can be achieved when        the dielectric permittivity ∈ is large and the azimuthal field        variations are slow (namely of small principal number m).        Material absorption is related to the material loss tangent:        Q_(abs) ∪Re{∈}/Im{∈}. Mode-solving calculations for this type of        disk resonances were performed using two independent methods:        numerically, 2D finite-difference frequency-domain (FDFD)        simulations (which solve Maxwell's Equations in frequency domain        exactly apart for spatial discretization) were conducted with a        resolution of 30pts/r; analytically, standard separation of        variables (SV) in polar coordinates was used.

TABLE 7 single disk λ/r Q^(abs) Q^(rad) Q Re{ε} = 147.7, m = 2 20.01(20.00) 10103 1988 (1992) 1661 (10075) (1663) Re{ε} = 65.6, m = 3 9.952(9.950) 10098 9078 (9168) 4780 (10087) (4802)

The results for two TE-polarized dielectric-disk subwavelength modes ofλ/r≧10 are presented in Table 7. Table 7 shows numerical FDFD (and inparentheses analytical SV) results for the wavelength and absorption,radiation and total loss rates, for two different cases ofsubwavelength-disk resonant modes. Note that disk- material loss-tangentIm{∈}/Re{∈}=10⁻⁴ was used. (The specific parameters corresponding to theplot in FIG. 15( a) are highlighted with bold in the table.) The twomethods have excellent agreement and imply that for a properly designedresonant low-loss-dielectric object values of Q_(rad)≧2000 andQ_(abs)˜10000 are achievable. Note that for the 3D case thecomputational complexity would be immensely increased, while the physicswould not be significantly different. For example, a spherical object of∈=147.7 has a whispering gallery mode with m=2, Q_(rad)=13962, andλ/r=17.

The required values of ∈, shown in Table 7, might at first seemunrealistically large. However, not only are there in the microwaveregime (appropriate for approximately meter-range coupling applications)many materials that have both reasonably high enough dielectricconstants and low losses (e.g. Titania, Barium tetratitanate, Lithiumtantalite etc.), but also ∈ could signify instead the effective index ofother known subwavelength surface-wave systems, such as surface modes onsurfaces of metallic materials or plasmonic (metal-like, negative-∈)materials or metallo-dielectric photonic crystals or plasmono-dielectricphotonic crystals.

To calculate now the achievable rate of energy transfer between twodisks 1 and 2, as shown in FIG. 15( b) we place them at distance Dbetween their centers. Numerically, the FDFD mode-solver simulationsgive κ through the frequency splitting of the normal modes of thecombined system (δ_(E)=2κ from Eq. (4)), which are even and oddsuperpositions of the initial single-disk modes; analytically, using theexpressions for the separation-of-variables eigenfields E_(1,2(r)) CMTgives κ throughκ=ω₁/2·∫d ³ r∈ ₂(r)E ₂*(r)E ₁(r)/∫d ³ r∈(r)|E ₁(r)|²,where ∈_(j)(r) and ∈(r) are the dielectric functions that describe onlythe disk j (minus the constant ∈_(o) background) and the whole spacerespectively. Then, for medium distances D/r=10−3 and for non-radiativecoupling such that D<2r_(c), where r_(c)=mλ/2π is the radius of theradiation caustic, the two methods agree very well, and we finally find,as shown in Table 8, strong-coupling factors in the range U˜1-50. Thus,for the analyzed examples, the achieved figure-of-merit values are largeenough to be useful for typical applications, as discussed below.

TABLE 8 two disks D/r Q^(rad) Q = ω/2Γ ω/2κ κ/Γ Re{ε} =147.7, 3 24781989 46.5(47.5) 42.4 (35.0) m = 2 5 2411 1946 298.0 6.5 (5.6) λ/r = 20(298.0) Q^(abs) ≈ 10093 7 2196 1804 769.7 2.3(2.2) (770.2) 10 2017 16811714 (1601) 0.98 (1.04) Re{ε}= 65.6, m = 3 3 7972 4455  144(140)30.9(34.3) λ/r ≈ 10 5 9240 4824 2242 (2083) 2.2(2.3) Q^(abs) ≈ 10096 79187 4810 7485 (7417) 0.64 (0.65)

Note that even though particular examples are presented and analyzedabove as examples of systems that use resonant electromagnetic couplingfor wireless energy transfer, those of self-resonant conducting coils,capacitively-loaded resonant conducting coils, inductively-loadedresonant conducting rods and resonant dielectric disks, any system thatsupports an electromagnetic mode with its electromagnetic energyextending much further than its size can be used for transferringenergy. For example, there can be many abstract geometries withdistributed capacitances and inductances that support the desired kindof resonances. In some examples, the resonant structure can be adielectric sphere. In any one of these geometries, one can choosecertain parameters to increase and/or optimize U or, if the Q's arelimited by external factors, to increase and/or optimize for k or, ifother system performance parameters are of importance, to optimizethose.

Illustrative Example

In one example, consider a case of wireless energy transfer between twoidentical resonant conducting loops, labeled by L₁ and L₂. The loops arecapacitively-loaded and couple inductively via their mutual inductance.Let r_(A) denote the loops' radii, N_(A) their numbers of turns, andb_(A) the radii of the wires making the loops. We also denote by D₁₂ thecenter-to-center separation between the loops.

Resonant objects of this type have two main loss mechanisms: ohmicabsorption and far-field radiation. Using the same theoretical method asin previous sections, we find that for r_(A)=7 cm, b_(A)=6 mm, andN_(A)=15 turns, the quality factors for absorption, radiation and totalloss are respectively, Q_(A,abs)=πf/δ_(A,abs)=3.19×10⁴,Q_(A,rad)=πf/Γ_(A,rad)=2.6×10⁸ and Q_(A)=πf/Γ_(A)=2.84×10⁴ at a resonantfrequency f=1.8×10⁷ Hz (remember that L₁ and L₂ are identical and havethe same properties). Γ_(A,abs), Γ_(A,rad) are respectively the rates ofabsorptive and radiative loss of L₁ and L₂, and the rate of couplingbetween L₁ and L₂ is denoted by κ₁₂.

When the loops are in fixed distinct parallel planes separated byD₁₂=1.4 m and have their centers on an axis (C) perpendicular to theirplanes, as shown in FIG. 17 a(Left), the coupling factor for inductivecoupling (ignoring retardation effects) is k₁₂≡κ₁₂/πf=7.68×10⁻⁵,independent of time, and thus the strong-coupling factor isU₁₂≡k₁₂Q_(A)=2.18. This configuration of parallel loops corresponds tothe largest possible coupling rate κ₁₂ at the particular separation D₁₂.

We find that the energy transferred to L₂ is maximum at timeT_(*)=κt_(*)=tan⁻¹(2.18)=1.14

t_(*)=4775(1/f) from Eq. (5), and constitutes η_(E*)=29% of the initialtotal energy from Eq. (6a), as shown in FIG. 17 a(Right). The energiesradiated and absorbed are respectively η_(rad,E)(t_(*))=7.2% andη_(abs,E)(t_(*))=58.1% of the initial total energy, with 5.8% of theenergy remaining in L₁.

We would like to be able to further increase the efficiency of energytransfer between these two resonant objects at their distance D₁₂. Insome examples, one can use an intermediate resonator between the sourceand device resonators, so that energy is transferred more efficientlyfrom the source to the device resonator indirectly through theintermediate resonator.

3. Efficient Energy-Transfer by a Chain of Three ‘Strongly Coupled’Resonances

FIG. 16 shows a schematic that generally describes one example of theinvention, in which energy is transferred wirelessly among threeresonant objects. Referring to FIG. 16, energy is transferred, over adistance D_(1B), from a resonant source object R₁ of characteristic sizer₁ to a resonant intermediate object R_(B) of characteristic size r_(B),and then, over an additional distance D_(B2), from the resonantintermediate object R_(B) to a resonant device object R₂ ofcharacteristic size r₂, where D_(1B)+D_(B2)=D. As described above, thesource object R₁ can be supplied power from a power generator that iscoupled to the source object R₁. In some examples, the power generatorcan be wirelessly, e.g., inductively, coupled to the source object R₁.In some examples, the power generator can be connected to the sourceobject R₁ by means of a wire or cable. Also, the device object R₂ can beconnected to a power load that consumes energy transferred to the deviceobject R₂. For example, the device object can be connected to e.g. aresistor, a battery, or other device. All objects are resonant objects.The wireless near-field energy transfer is performed using the field(e.g. the electromagnetic field or acoustic field) of the system ofthree resonant objects.

Different temporal schemes can be employed, depending on theapplication, to transfer energy among three resonant objects. Here wewill consider a particularly simple but important scheme: a one-timefinite-amount energy-transfer scheme

3.1 Finite-Amount Energy-Transfer Efficiency

Let again the source, intermediate and device objects be 1, B, 2respectively and their resonance modes, which we will use for the energyexchange, have angular frequencies ω_(1,B,2), frequency-widths due tointrinsic (absorption, radiation etc.) losses Γ_(1,B,2) and (generally)vector fields F_(1,B,2)(r), normalized to unity energy. Once the threeresonant objects are brought in proximity, they can interact and anappropriate analytical framework for modeling this resonant interactionis again that of the well-known coupled-mode theory (CMT), which cangive a good description of the system for resonances having qualityfactors of at least, for example, 10. Then, using e^(−iωt) timedependence, for the chain arrangement shown in FIG. 16, the fieldamplitudes can be shown to satisfy, to lowest order:

$\begin{matrix}{\mspace{79mu}{{{\frac{\mathbb{d}\;}{\mathbb{d}t}{a_{1}(t)}} = {{{- {{\mathbb{i}}\left( {\omega_{1} - {{\mathbb{i}}\;\Gamma_{1}}} \right)}}{a_{1}(t)}} + {{\mathbb{i}}\;\kappa_{11}{a_{1}(t)}} + {{\mathbb{i}}\;\kappa_{1\; B}{a_{B}(t)}}}}{{\frac{\mathbb{d}\;}{\mathbb{d}t}{a_{B}(t)}} = {{{- {{\mathbb{i}}\left( {\omega_{B} - {{\mathbb{i}}\;\Gamma_{B}}} \right)}}{a_{B}(t)}} + {{\mathbb{i}}\;\kappa_{BB}{a_{B}(t)}} + {{\mathbb{i}}\;\kappa_{B\; 1}{a_{1}(t)}} + {{\mathbb{i}}\;\kappa_{B\; 2}{a_{2}(t)}}}}\mspace{79mu}{{\frac{\mathbb{d}\;}{\mathbb{d}t}{a_{2}(t)}} = {{{- {{\mathbb{i}}\left( {\omega_{2} - {{\mathbb{i}}\;\Gamma_{2}}} \right)}}{a_{2}(t)}} + {{\mathbb{i}}\;\kappa_{22}{a_{2}(t)}} + {{\mathbb{i}}\;\kappa_{2\; B}{a_{B}(t)}}}}}} & (34)\end{matrix}$where κ_(11,BB,22) are the shifts in each object's frequency due to thepresence of the other, which are a second-order correction and can beabsorbed into the eigenfrequencies by settingκ_(1,B,2)→ω_(1,B,2)+κ_(11,BB,22), and κ_(ij) are the couplingcoefficients, which from the reciprocity requirement of the system mustsatisfy κ_(ij)=κ_(ji). Note that, in some examples, the direct couplingcoefficient κ₁₂ between the resonant objects 1 and 2 may be much smallerthan the coupling coefficients κ_(1B) and κ_(B2) between these tworesonant objects with the intermediate object B, implying that thedirect energy transfer between 1 and 2 is substantially dominated by theindirect energy transfer through the intermediate object. In someexamples, if the coupling rates κ_(1B) and κ_(B2) are at least 5 timeslarger than the direct coupling rate κ₁₂, using an intermediateresonator may lead to an improvement in terms of energy transferefficiency, as the indirect transfer may dominate the direct transfer.Therefore, in the CMT Eqs. (34) above, the direct coupling coefficientκ₁₂ has been ignored, to analyze those particular examples.

The three resonant modes of the combined system are found bysubstituting [a₁(t) a_(B)(t), a₂ (t)]=[A₁,A_(B),A₂]e^(−i ωt). When theresonators 1 and 2 are at exact resonance ω₁=ω₂=ω_(A) and forΓ₁=Γ₂=Γ_(A), the resonant modes have complex resonant frequenciesω _(±)=ω_(AB)±√{square root over ((Δω_(AB))²+κ_(1B) ²+κ_(B2) ²)} and ω_(ds)=ω_(A) −iΓ _(A)where ω_(AB)=[(ω_(A)+ω_(B))−i(Γ_(A)+Γ_(B))]/2,Δω_(AB)=[(ω_(A)−ω_(B))−i(Γ_(A)−Γ_(B))]/2 and whose splitting we denoteas {tilde over (δ)}_(E)≡ ω ₊− ω ⁻, and corresponding resonant fieldamplitudes

$\begin{matrix}{{{\overset{\rightarrow}{V}}_{\pm} = {\begin{bmatrix}A_{1} \\A_{B} \\A_{2}\end{bmatrix}_{\pm} = \begin{bmatrix}\kappa_{1\; B} \\{{\Delta\;\omega_{AB}} \mp \sqrt{\left( {\Delta\;\omega_{AB}} \right)^{2} + \kappa_{1\; B}^{2} + \kappa_{B\; 2}^{2}}} \\\kappa_{B\; 2}\end{bmatrix}}}{and}{{\overset{\rightarrow}{V}}_{ds} = {\begin{bmatrix}A_{1} \\A_{B} \\A_{2}\end{bmatrix}_{ds} = \begin{bmatrix}{- \kappa_{B\; 2}} \\0 \\\kappa_{1\; B}\end{bmatrix}}}} & \left( {35\; b} \right)\end{matrix}$Note that, when all resonators are at exact resonanceω₁=ω₂(=ω_(A))=ω_(B) and for Γ₁=Γ₂(=Γ_(A))=Γ_(B), we get Δω_(AB)=0,{tilde over (δ)}_(E)=2√{square root over (κ_(1B) ²+κ_(B2) ²)}, and then

$\begin{matrix}{{{\overset{\_}{\omega}}_{\pm} = {{\omega_{A} \pm \sqrt{\kappa_{1\; B}^{2} + \kappa_{B\; 2}^{2}}} - {{\mathbb{i}}\;\Gamma_{A}}}}{and}{{\overset{\_}{\omega}}_{ds} = {\omega_{A} - {{\mathbb{i}}\;\Gamma_{A}}}}} & \left( {36\; a} \right) \\{{{\overset{\rightarrow}{V}}_{\pm} = {\begin{bmatrix}A_{1} \\A_{B} \\A_{2}\end{bmatrix}_{\pm} = \begin{bmatrix}\kappa_{1\; B} \\{\mp \sqrt{\kappa_{1\; B}^{2} + \kappa_{B\; 2}^{2}}} \\\kappa_{B\; 2}\end{bmatrix}}}{and}{{{\overset{\rightarrow}{V}}_{ds} = {\begin{bmatrix}A_{1} \\A_{B} \\A_{2}\end{bmatrix}_{ds} = \begin{bmatrix}{- \kappa_{B\; 2}} \\0 \\\kappa_{1\; B}\end{bmatrix}}},}} & \left( {36\; b} \right)\end{matrix}$namely we get that the resonant modes split to a lower frequency, ahigher frequency and a same frequency mode.

Assume now that at time t=0 the source object 1 has finite energy|a₁(0)|², while the intermediate and device objects have|a_(B)(0)|²=|a₂(0)|²=0. Since the objects are coupled, energy will betransferred from 1 to B and from B to 2. With these initial conditions,Eqs. (34) can be solved, predicting the evolution of thefield-amplitudes. The energy-transfer efficiency will be {tilde over(η)}_(E)≡|a₂(t)|²/|a₁(0)|². The ratio of energy converted to loss due toa specific loss mechanism in resonators 1, B and 2, with respective lossrates Γ_(1,loss), Γ_(B,loss) and Γ_(2,loss) will be {tilde over(η)}_(loss,E)=∫₀^(t)dτ[2Γ_(1,loss)|a₁(τ)|²+2Γ_(B,loss)|a_(B)(τ)|²+2Γ_(2,loss)|a₂(τ)|²]/|a₁(0)|².At exact resonance ω₁=ω₂(=ω_(A))=ω_(B) (an optimal condition) and in thespecial symmetric case Γ₁=Γ₂=Γ_(A) and κ_(1B)=κ_(B2)=κ, the fieldamplitudes are

$\begin{matrix}{{a_{1}\left( \overset{\sim}{T} \right)} = {\frac{1}{2}{\mathbb{e}}^{{- {\mathbb{i}}}\;\omega_{A}t}{{\mathbb{e}}^{{- \overset{\sim}{T}}/\overset{\sim}{U}}\left\lbrack {{\overset{\sim}{\Delta}\frac{\sin\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)}{\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}}} + {\cos\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)} + {\mathbb{e}}^{{- \overset{\sim}{\Delta}}\;\overset{\sim}{T}}} \right\rbrack}}} & \left( {37\; a} \right) \\{\mspace{79mu}{{a_{B}\left( \overset{\sim}{T} \right)} = {\frac{1}{2}{\mathbb{e}}^{{- {\mathbb{i}}}\;\omega_{A}t}{\mathbb{e}}^{{- \overset{\sim}{T}}/\overset{\sim}{U}}\frac{\sin\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)}{\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}}}}} & \left( {37\; b} \right) \\{{a_{2}\left( \overset{\sim}{T} \right)} = {\frac{1}{2}{\mathbb{e}}^{{- {\mathbb{i}}}\;\omega_{A}t}{{\mathbb{e}}^{{- \overset{\sim}{T}}/\overset{\sim}{U}}\left\lbrack {{\overset{\sim}{\Delta}\frac{\sin\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)}{\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}}} + {\cos\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)} - {\mathbb{e}}^{{- \overset{\sim}{\Delta}}\;\overset{\sim}{T}}} \right\rbrack}}} & \left( {37\; c} \right)\end{matrix}$where {tilde over (T)}≡√{square root over (2)}κt, {tilde over(Δ)}⁻¹=2√{square root over (2)}κ/(Γ_(B)−Γ_(A)) and Ũ=2√{square root over(2)}κ/(Γ_(A)+Γ_(B)).

In some examples, the system designer can adjust the duration of thecoupling t at will. In some examples, the duration t can be adjusted tomaximize the device energy (and thus efficiency {tilde over (η)}_(E)).Then, in the special case above, it can be inferred from Eq. (37c) that{tilde over (Γ)}_(E) is maximized for the {tilde over (T)}={tilde over(T)}_(*) that satisfies

$\begin{matrix}{{{\left\lbrack {\overset{\sim}{\Delta} - {\overset{\sim}{U}\left( {1 - {\overset{\sim}{\Delta}}^{2}} \right)}} \right\rbrack\frac{\sin\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)}{\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}}} + {\left( {1 - {\overset{\sim}{\Delta}\;\overset{\sim}{U}}} \right)\left\lbrack {{\cos\left( {\sqrt{1 - {\overset{\sim}{\Delta}}^{2}}\overset{\sim}{T}} \right)} - {\mathbb{e}}^{{- \overset{\sim}{\Delta}}\;\overset{\sim}{T}}} \right\rbrack}} = 0.} & (38)\end{matrix}$In general, this equation may not have an obvious analytical solution,but it does admit a simple solution in the following two cases:When Γ_(A)=Γ_(B)

{tilde over (Δ)}=0, Ũ=√{square root over (2)}κ/Γ_(B), the energytransfer from resonator 1 to resonator 2 is maximized at{tilde over (T)} _(*)({tilde over (Δ)}=0)=2 tan⁻¹ Ũ  (39)resulting in an energy-transfer efficiency

$\begin{matrix}{{{{\overset{\sim}{\eta}}_{E}\left( {{\overset{\sim}{T}}_{*},{\overset{\sim}{\Delta} = 0}} \right)} = \left\lbrack {\frac{{\overset{\sim}{U}}^{2}}{1 + {\overset{\sim}{U}}^{2}}{\exp\left( {- \frac{2\;\tan^{- 1}\overset{\sim}{U}}{\overset{\sim}{U}}} \right)}} \right\rbrack^{2}},} & (40)\end{matrix}$which has the form of the two-object energy-transfer efficiency of Eq.(6b) but squared.When Γ_(A)=0

{tilde over (Δ)}⁻¹=Ũ=2√{square root over (2)}κ/Γ_(B), the energytransfer from resonator 1 to resonator 2 is maximized at

$\begin{matrix}{{{\overset{\sim}{T}}_{*}\left( {{\overset{\sim}{\Delta}}^{- 1} = \overset{\sim}{U}} \right)} = \left\{ \begin{matrix}{{\pi\;{\overset{\sim}{U}/\sqrt{{\overset{\sim}{U}}^{2} - 1}}},{\overset{\sim}{U} > 1}} \\{\infty,{\overset{\sim}{U} \leq 1}}\end{matrix} \right.} & (41)\end{matrix}$resulting in an energy-transfer efficiency

$\begin{matrix}{{\eta_{E}\left( {{\overset{\sim}{T}}_{*},{{\overset{\sim}{\Delta}}^{- 1} = \overset{\sim}{U}}} \right)} = {\frac{1}{4} \cdot \left\{ \begin{matrix}{\left. {\left\lbrack {1 + \exp} \right)\left( {{- \pi}/\sqrt{{\overset{\sim}{U}}^{2} - 1}} \right)} \right\rbrack^{2},{\overset{\sim}{U} > 1}} \\{1,{\overset{\sim}{U} \leq 1.}}\end{matrix} \right.}} & (42)\end{matrix}$In both cases, and in general for any {tilde over (Δ)}, the efficiencyis an increasing function of Ũ. Therefore, once more resonators thathave high quality factors are preferred. In some examples, one maydesign resonators with Q>50. In some examples, one may design resonatorswith Q>100.Illustrative Example

Returning to our illustrative example, in order to improve the ˜29%efficiency of the energy transfer from L₁ to L₂, while keeping thedistance D₁₂ separating them fixed, we propose to introduce anintermediate resonant object that couples strongly to both L₁ and L₂,while having the same resonant frequency as both of them. In oneexample, we take that mediator to also be a capacitively-loadedconducting-wire loop, which we label by L_(B). We place L_(B) at equaldistance (D_(1B)=D_(B2)=D₁₂/2=0.7 m) from L₁ and L₂ such that its axisalso lies on the same axis (C), and we orient it such that its plane isparallel to the planes of L₁ and L₂, which is the optimal orientation interms of coupling. A schematic diagram of the three-loops configurationis depicted in FIG. 17 b(Left).

In order for L_(B) to couple strongly to L₁ and L₂, its size needs to besubstantially larger than the size of L₁ and L₂. However, this increasein the size of L_(B) has considerable drawback in the sense that it mayalso be accompanied by significant decrease in its radiation qualityfactor. This feature may often occur for the resonant systems of thistype: stronger coupling can often be enabled by increasing the objects'size, but it may imply stronger radiation from the object in question.Large radiation may often be undesirable, because it could lead tofar-field interference with other RF systems, and in some systems alsobecause of safety concerns. For r_(B)=70 cm, b_(B)=1.5 cm, and N_(B)=1turn, we get Q_(B,abs)=πf/Γ_(B,abs)=7706, Q_(B,rad)=πf/Γ_(B,rad)=400, soQ_(B)=πf/Γ_(B)=380, and k_(1B)≡κ_(1B)πf=k_(B2)=0.0056, so Ũ=2√{squareroot over (2)}κ/(Γ_(A)+Γ_(B))=5.94 and {tilde over (Δ)}⁻¹=2√{square rootover (2)}κ/(Γ_(B)−Γ_(A))=6.1, at f=1.8×10⁷ Hz. Note that since thecoupling rates κ_(1B) and κ_(B2) are ≈70 times larger than κ₁₂, indeedwe can ignore the direct coupling between L₁ and L₂, and focus only onthe indirect energy transfer through the intermediate loop L_(B),therefore the analysis of the previous section can be used.

The optimum in energy transferred to L₂ occurs at time {tilde over(T)}_(*)=√{square root over (2)}κt_(*)=3.21

t_(*)=129(1/f), calculated from Eq. (38), and is equal to {tilde over(η)}_(E*)=61.5% of the initial energy from Eq. (37c). [Note that, sinceQ_(A)

Q_(B), we could have used the analytical conclusions of the case {tildeover (Δ)}⁻¹=Ũ and then we would have gotten a very close approximationof {tilde over (T)}_(*)=3.19 from Eq. (41).] The energy radiated is{tilde over (η)}_(rad,E)(t_(*))=31.1%, while the energy absorbed is{tilde over (η)}_(abs,E)(t_(*))=3.3%, and 4.1% of the initial energy isleft in L₁. Thus, the energy transferred, now indirectly, from L₁ to L₂has increased by factor of 2 relative to the two-loops direct transfercase. Furthermore, the transfer time in the three-loops case is now ≈35times shorter than in the two-loops direct transfer, because of thestronger coupling rate. The dynamics of the energy transfer in thethree-loops case is shown in FIG. 17 b(Right).

Note that the energy radiated in the three-loop system has undesirablyincreased by factor of 4 compared to the two-loop system. We would liketo be able to achieve a similar improvement in energy-transferefficiency, while not allowing the radiated energy to increase. In thisspecification, we disclose that, in some examples, this can be achievedby appropriately varying the values of the coupling strengths betweenthe three resonators.

4. Efficient Energy-Transfer by a Chain of Three Resonances withAdiabatically Varying Coupling Strengths

Consider again the system of a source resonator R₁, a device resonatorR₂ and an intermediate resonator R_(B). For the purposes of the presentanalysis, R₁ and R₂ will be assumed to have negligible mutualinteractions with each other, while each of them can be strongly coupledto R_(B), with coupling rates κ_(1B) and κ_(B2) respectively. Note that,in some examples of wireless energy transfer systems, for the resonantobject R_(B) to be able to have strong coupling with other resonantobjects, it may be accompanied with inferior loss properties compared toR₁ and R₂, usually in terms of substantially larger radiation losses.

It was seen in a previous section that, when the resonators 1 and 2 areat exact resonance ω_(l)=ω₂=ω_(A) and for Γ₁=Γ₂=Γ_(A), the systemsupports a resonant state (eigenstate) with resonant frequency(eigenfrequency) ω _(ds)=ω_(A)−iΓ_(A) and resonant field amplitudes{right arrow over (V)}_(ds)=[−κ_(B2) 0 κ_(1B)]^(T)/√{square root over(κ_(1B) ²+κ_(B2) ²)}, where here we normalized it. This eigenstate{right arrow over (V)}_(ds) we call the “dark state” in analogy withatomic systems and the related phenomenon of Electromagnetically InducedTransparency (EIT), wherein complete population transfer between twoquantum states through a third lossy state, coupled to each of the othertwo states, is enabled. The dark state is the most essential buildingblock of our proposed efficient weakly-radiative energy-transfer scheme,because it has no energy at all in the intermediate (lossy) resonatorR_(B), i.e. a_(B)(t)=0∀t whenever the three-object system is in state{right arrow over (V)}_(ds). In fact, if Γ_(A)→0, then this state iscompletely lossless, or if Γ_(A,rad)→0, then this state is completelynon-radiative. Therefore, we disclose using predominantly this state toimplement the wireless energy transfer. By doing that, the proposedenergy transfer scheme can be made completely lossless, in the limitΓ_(A)→0, no matter how large is the loss rate Γ_(B) as shown in FIG. 20,or completely non-radiative, in the limit Γ_(A,rad)→0, no matter howlarge is the radiative loss rate Γ_(B,rad). Since the energy transferefficiency increases as the quality factors of the first (source) andsecond (device) resonances increase, one may design these resonators tohave a high quality factor. In some examples, one may design resonatorswith Q_(A)>50. In some examples, one may design resonators withQ_(A)>100.

Let us demonstrate how it is possible to use the dark state for energytransfer with minimal loss. From the expression of {right arrow over(V)}_(ds) one can see that, if the three-object system is in the state{right arrow over (V)}_(ds), then, in general, there is energy in thesource resonator with field amplitude proportional to the coupling rateκ_(B2) between the device resonator and the intermediate resonator, andthere is also energy in the device resonator with field amplitudeproportional to the coupling rate κ_(1B) between the source resonatorand the intermediate resonator. Then, κ_(1B)=0 corresponds to all thesystem's energy being in R₁, while κ_(B2)=0 corresponds to all thesystem's energy being in R₂.

So, the important considerations necessary to achieve efficientweakly-radiative energy transfer, consist of preparing the systeminitially in state {right arrow over (V)}_(ds) and varying the couplingrates in time appropriately to evolve this state {right arrow over(V)}_(ds) in a way that will cause energy transfer. Thus, if at t=0 allthe energy is in R₁, then one should have κ_(1B)(t=0)<<κ_(B2)(t=0), forexample κ_(1B)(t=0)=0 and κ_(B2)(t=0)≠0. In order for the total energyof the system to end up in R₂, at a time t_(EIT) when the full variationof the coupling rates has been completed, we should haveκ_(1B)(t=t_(EIT))>>κ_(B2)(t=t_(EIT)), for example κ_(1B)(t=t_(EIT))≠0and κ_(B2)(t=t_(EIT))=0. This ensures that the initial and final statesof the three-object system are parallel to {right arrow over (V)}_(ds).However, a second important consideration is to keep the three-objectsystem at all times in {right arrow over (V)}_(ds)(t), even as κ_(1B)(t)and κ_(B2)(t) are varied in time. This is crucial in order to preventthe system's energy from getting into any of the two other eigenstates{right arrow over (V)}_(±) and thus getting into the intermediate objectR_(B), which may be highly radiative or lossy in general, as in theexample of FIG. 17. This consideration requires changing κ_(1B)(t) andκ_(B2)(t) slowly enough so as to make the entire three-object systemadiabatically follow the time evolution of {right arrow over (V)}_(ds)(t). The criterion for adiabatic following can be expressed, in analogyto the population transfer case as

$\begin{matrix}{{\left\langle {{\overset{\rightarrow}{V}}_{\pm}❘\frac{\mathbb{d}{\overset{\rightarrow}{V}}_{ds}}{\mathbb{d}t}} \right\rangle }{{{{\overset{\_}{\omega}}_{\pm} - {\overset{\_}{\omega}}_{ds}}}}} & (43)\end{matrix}$where {right arrow over (V)}_(±) are the remaining two eigenstates ofthe system, with corresponding eigenvalues ω _(±), as shown earlier.Note that any functional dependence of the coupling rates with time(with duration parameter t_(EIT)) will work, provided it satisfies theadiabaticity criterion Eq. (43) above. The time functional can belinear, sinusoidal (as in the illustrative example to follow) or thetemporal analog of a Butterworth (maximally flat) taper, a Chebyshevtaper, an exponential taper and the like.

Referring to FIG. 18, an example coupling rate adjustment system 300 foradjusting coupling rates for the one or more of the resonator structuresR₁, R₂, or R_(B) is shown. As described, the coupling rates between thefirst and intermediate resonator structures and the intermediate andsecond resonator structures are characterized by κ_(1B) and κ_(B2)respectively. These coupling rates, κ_(1B) and κ_(B2), are several times(e.g., 70 times) greater than the coupling rate κ₁₂ between the firstand second resonator structure. In some examples, the coupling rateadjustment system can be a mechanical, electrical, or electro-mechanicalsystem 300 for dynamically adjusting, e.g., rotating, or effecting atranslational movement (X, Y, Z), of the one or more resonatorstructures with respect to each other.

In some examples, the coupling rate κ_(1B) is much smaller than thecoupling rate the coupling rate κ_(B2) at the beginning of the energytransfer. By the end, i.e., when a substantial amount of energy has beentransferred from the first resonator structure R₁ to the secondresonator structure, R₂, the coupling rate κ_(1B) is much greater thanthe coupling rate κ_(B2). In some examples, the coupling rate κ_(1B) canbe set to a fixed value while the coupling rate κ_(B2) is being variedfrom its maximum to its minimum value. In some examples, the couplingrate κ_(B2) can be set to a fixed value while the coupling rate κ_(1B)is being varied from its minimum to its maximum value. In some examples,the coupling rate κ_(1B) can be varied from a minimum to a maximum valuewhile the coupling rate κ_(B2) is being varied from its maximum tominimum value.

Referring now to FIG. 19, a graph for implementing an example couplingrate adjustment system 300 is shown. As shown, in some examples, thecoupling rate κ_(1B) is set at its minimum value at time, t=0, andincreased as a function of time (see, for example, equation 44), whilethe coupling rate κ_(B2) is at its maximum value at t=0 and decreased asa function of time (see, for example, equation 45). Accordingly, at thebeginning (t=0), the value of κ_(1B) is much smaller than the value ofκ_(B2). In some examples, the value of κ_(1B) can be selected to be anyvalue much smaller than the value of κ_(B2) During the wireless energytransfer, the value of κ_(1B) is increased, while the value of κ_(B2) isdecreased. After a predetermined amount of time t_(EIT) has elapsed(e.g., after a substantial amount of energy has been transferred to thesecond resonator), the value of κ_(1B) becomes much greater than thevalue of κ_(B2).

In some implementations, the coupling rate adjustment system 300 caneffect an adjustment of coupling rates between the resonator structuresby changing a relative orientation of one or more of the resonatorstructures with respect to each other. For example, referring again toFIG. 18, the first and second resonator structures, R₁ and R₂, can berotated about their respective axes (e.g., varying angles θ₁ and θ₂),with respect to the intermediate resonator structure R_(B) tosimultaneously change κ_(1B) and κ_(B2). Alternatively, the orientationof the intermediate resonator structure, R_(B), can be adjusted, e.g.,rotated about an axis, with respect to the first and second resonatorstructures to simultaneously change κ_(1B) and κ_(B2). Alternatively,the orientation of only the first resonator structure R₁ can be rotatedabout its respective axis to change κ_(1B), while the orientations of R₂and R_(B) are fixed and thus κ_(B2) is fixed to a value intermediatebetween the minimum and maximum values of κ_(1B). Alternatively, theorientation of only the second resonator structure R₂ can be rotatedabout its respective axis to change κ_(B2), while the orientations of R₁and R_(B) are fixed and thus κ_(1B) is fixed to a value intermediatebetween the minimum and maximum values of κ_(B2).

In some implementations, the coupling rate adjustment system 300 caneffect an adjustment of coupling rates between the resonator structuresby translationally moving one or more of the resonator structures withrespect to each other. For example, the positions of the first andsecond resonator structures, R₁ and R₂, can be adjusted, e.g., movedalong an axis, with respect to the intermediate resonator structureR_(B) to simultaneously change κ_(1B) and κ_(B2). Alternatively, aposition of the intermediate resonator structure, R_(B), can beadjusted, e.g., moved along an axis, with respect to the first andsecond resonator structures to simultaneously change κ_(1B) and κ_(B2).Alternatively, a position of only the first resonator structure, R₁, canbe adjusted, e.g., moved along an axis, with respect to the intermediateR_(B) resonator structure to change κ_(1B), while the positions of R₂and R_(B) are fixed and thus κ_(B2) is fixed to a value intermediatebetween the minimum and maximum values of κ_(1B). Alternatively, aposition of only the second resonator structure, R₂, can be adjusted,e.g., moved along an axis, with respect to the intermediate R_(B)resonator structure to change κ_(B2), while the positions of R₁ andR_(B) are fixed and thus κ_(1B) is fixed to a value intermediate betweenthe minimum and maximum values of κ_(B2).

In some examples, the coupling rate adjustment system 300 candynamically adjust an effective size of the resonator objects to effectadjustments in the coupling rates κ_(1B) and κ_(B2) similar to thatdescribed above. The effective size can be adjusted by changing aphysical size of the resonator objects. For example, the physical sizecan be adjusted by effecting mechanical changes in area, length, orother physical aspect of one or more of the resonator structures. Theeffective size can also be adjusted through non-mechanical changes, suchas, but not limited to, applying a magnetic field to change thepermeability of the one or more of the resonator structures.

In principle, one would think of making the transfer time t_(EIT) aslong as possible to ensure adiabaticity. However there is limitation onhow slow the transfer process can optimally be, imposed by the losses inR₁ and R₂. Such a limitation may not be a strong concern in typicalatomic EIT case, because the initial and final states there can bechosen to be non-lossy ground states. However, in our case, losses in R₁and R₂ are not avoidable, and can be detrimental to the energy transferprocess whenever the transfer time t_(EIT) is not less than 1/Γ_(A).This is because, even if the three-object system is carefully kept in{right arrow over (V)}_(ds) at all times, the total energy of the systemwill decrease from its initial value as a consequence of losses in R₁and R₂. Thus the duration of the transfer may be a compromise betweenthese two limits: the desire to keep t_(EIT) long enough to ensurenear-adiabaticity, but short enough not to suffer from losses in R₁ andR₂.

Given a particular functional variation of the coupling rates with timewith variation duration parameter t_(EIT), one may calculate the optimalenergy transfer efficiency in the following way: First, for eacht_(EIT), determine the optimal time t_(*), at which the energy at R₂ ismaximized and the transfer process may be terminated. Then find theoptimal variation time t_(EIT*) based on the counteracting mechanismsdiscussed above. The optimal efficiency of energy transfer {tilde over(η)}_(EIT,E*) can then be calculated. In most cases, this procedure mayneed to be done numerically using the CMT Eqs. (34) as analyticalsolutions may not be possible. With respect to optimizing the functionaldependence of the coupling rates with time, one may choose one thatminimizes the coupling of energy to the eigenstates {right arrow over(V)}_(±) for a given t_(EIT), which may lead to the temporal analog of aChebyshev taper.

In some examples, the optimal t_(EIT) may not be long enough for theadiabadicity criterion of Eq. (43) to be always satisfied. In thosecases, some energy may get into at least one of the lossy states {rightarrow over (V)}_(±). Still significant improvement in efficiency andradiation loss may be achieved by the mode of operation where thecoupling rates are variable, compared to the mode of operation where thecoupling rates are constant, provided the maximum energy that enters thestates {right arrow over (V)}_(±) is much less in the variable ratescenario than in the constant rate scenario. In examples, using theproposed scheme of time-varying coupling rates may be advantageous aslong as the maximum energy stored in the intermediate resonator issubstantially small. In some examples, substantially small may be atmost 5% of the peak total energy of the system. In some examples,substantially small may be at most 10% of the peak total energy of thesystem.

We can now also see why the mode of operation of the system where thecoupling rates are kept constant in time may cause a considerable amountof lost (and especially radiated) energy, compared to the proposed modeof operation where the coupling rates are varied adiabatically in time.The reason is that, when κ_(1B)=κ_(B2)=const, the energies in R₁ and R₂will always be equal to each other if the three-object system is to stayin {right arrow over (V)}_(as). So one cannot transfer energy from R₁ toR₂ by keeping the system purely in state {right arrow over (V)}_(ds);note that even the initial state of the system, in which all the energyis in R₁ and there is no energy in R₃, cannot be solely in {right arrowover (V)}_(ds) for fixed nonzero κ_(1B) and κ_(B2), and has nonzerocomponents along the eigenstates {right arrow over (V)}_(±) whichimplies a finite energy will build up in R_(B), and consequently resultin an increased radiation, especially if Γ_(B,rad)

Γ_(A,rad), which may be the case if the resonator R_(B) is chosen largeenough to couple strongly to R₁ and R₂, as explained earlier.

Illustrative Example

The previous analysis explains why a considerable amount of energy wasradiated when the inductive coupling rates of the loops were keptconstant in time, like in FIG. 17 b. Let us now consider themodifications necessary for an adiabatically-varied-K three-loopsindirect transfer scheme, as suggested in the previous section, aimingto reduce the total radiated energy back to its reasonable value in thetwo-loops direct transfer case (FIG. 17 a), while maintaining the totalenergy transfer at level comparable to the constant-κ three-loopsindirect transfer case (FIG. 17 b). In one example, shown in FIG. 17c(Left and Right), we will keep the orientation of L_(B) fixed, andstart initially (t=0) with L₁ perpendicular to L_(B) for κ_(1B)=0 and L₂parallel to L_(B) for κ_(B2)=max, then uniformly rotate L₁ and L₂, atthe same rates, until finally, at (t=t_(EIT)), L₁ becomes parallel toL_(B) for κ_(1B)=max and L₂ perpendicular to L_(B) for κ_(B2)=0, wherewe stop the rotation process. In this example, we choose a sinusoidaltemporal variation of the coupling rates:κ_(1B)(t)=κ sin(πt/2t _(EIT))  (44)κ_(B2)(t)=κ cos(πt/2t _(EIT))  (45)for 0<t<t_(EIT), and k_(1B)≡κ_(1B)/πf=k_(B2)=0.0056 as before. By usingthe same CMT analysis as in Eq. (34), we find, in FIG. 17 c(Center),that for an optimal t_(EIT*)=1989(1/f), an optimum transfer of {tildeover (η)}_(EIT,E*)=61.2% can be achieved at t_(*)=1796(1/f), with only8.2% of the initial energy being radiated, 28.6% absorbed, and 2% leftin L₁. This is quite remarkable: by simply rotating the loops during thetransfer, the energy radiated has dropped by factor of 4, while keepingthe same 61% level of the energy transferred.

This considerable decrease in radiation may seem surprising, because theintermediate resonator L_(B), which mediates all the energy transfer, ishighly radiative (≈650 times more radiative than L₁ and L₂), and thereis much more time to radiate, since the whole process lasts 14 timeslonger than in FIG. 17 b. Again, the clue to the physical mechanismbehind this surprising result can be obtained by observing thedifferences between the curves describing the energy in R_(B) in FIG. 17b and FIG. 17 c. Unlike the case of constant coupling rates, depicted inFIG. 17 b, where the amount of energy ultimately transferred to L₂ goesfirst through the intermediate loop L_(B), with peak energy storage inL_(B) as much as 30% of the peak total energy of the system, in the caseof time-varying coupling rates, shown in FIG. 17 c, there is almostlittle or no energy in L_(B) at all times during the transfer. In otherwords, the energy is transferred quite efficiently from L_(i) to L₂,mediated by L_(B) without considerable amount of energy ever being inthe highly radiative intermediate loop L_(B). (Note that direct transferfrom L₁ to L₂ is identically zero here since L₁ is always perpendicularto L₂, so all the energy transfer is indeed mediated through L_(B)). Insome examples, improvement in efficiency and/or radiated energy canstill have been accomplished if the energy transfer had been designedwith a time t_(EIT) smaller than its optimal value (perhaps to speed upthe process), if the maximum energy accumulated inside the intermediateresonator was less than 30%. For example, improvement can have beenachieved for maximum energy accumulation inside the intermediateresonator of 5% or even 10%.

An example implementation of the coupling rate adjustment system 300 isdescribed below, where the resonators are capacitively-loaded loops,which couple to each other inductively. At the beginning (t=0), thecoupling rate adjustment system 300 sets the relative orientation of thefirst resonator structure L₁ to be perpendicular to the intermediateresonator structure L_(B). At this orientation, the value of thecoupling rate κ_(1B) between the first and intermediate resonatorstructure is at its minimum value. Also, the coupling rate adjustmentsystem 300 can set the relative orientation of the second resonatorstructure L₂ to be parallel to the intermediate resonator structureL_(B). At this orientation, the value of the coupling rate κ_(B2) is ata maximum value. During wireless energy transfer, the coupling rateadjustment system 300 can effect the rotation of the first resonatorstructure L₁ about its axis so that the value of κ_(1B) is increased. Insome examples, the coupling rate adjustment system 300 can also effectthe rotation of the second resonator structure, L₂, about its axis sothat the value of κ_(B2) is decreased. In some examples, a similareffect can be achieved by fixing L₁ and L₂ to be perpendicular to eachother and rotating only L_(B) to be parallel to L₂ and perpendicular toL₁ at t=0 and parallel to L₁ and perpendicular to L₂ at t=t_(EIT). Insome examples, a similar effect can be achieved by fixing L_(B) and oneof L₁ and L₂ (e.g., L₁) at a predetermined orientation (e.g. at 45degrees with respect to the intermediate resonator L_(B)) and rotatingonly the other of L₁ and L₂ (e.g., L₂ from parallel to L_(B) at t=0 toperpendicular to L_(B) at t=t_(EIT)).

Similarly, in some implementations, at the beginning (t=0), the couplingrate adjustment system 300 can set the position of the first resonatorstructure L₁ at a first large predetermined distance from theintermediate resonator structure L_(B) so that the value of the couplingrate IC_(1B) is at its minimum value. Correspondingly, the coupling rateadjustment system 300 can set the position of the second resonatorstructure L₂ at a second small predetermined distance from theintermediate resonator structure L_(B) so that the value of the couplingrate κ_(B2) between the first and intermediate resonator structure is atits maximum value. During wireless energy transfer, the coupling rateadjustment system 300 can affect the position of the first resonatorstructure L₁ to bring it closer to L_(B) so that the value of κ_(1B) isincreased. In some examples, the coupling rate adjustment system 300 canalso effect the position of the second resonator structure, L₂, to takeit away from L_(B) so that the value of κ_(B2) is decreased. In someexamples, a similar effect can be achieved by fixing L₁ and L₂ to be ata fixed distance to each other and effecting the position of only L_(B)to be close to L₂ and away from L₁ at t=0 and close to L₁ and away fromL₂ at t=t_(EIT). In some examples, a similar effect can be achieved byfixing L_(B) and one of L₁ and L₂ (e.g., L₁) at a predetermined (not toolarge but not too small) distance and effecting the position only theother of L₁ and L₂ (e.g., L₂ from close to L_(B) at t=0 to away fromL_(B) at t=t_(EIT)).

5. Comparison of Static and Adiabatically Dynamic Systems

In the abstract case of energy transfer from R₁ to R₂, where noconstraints are imposed on the relative magnitude of κ, Γ_(rad) ^(A),Γ_(rad) ^(B), Γ_(abs) ^(A), and Γ_(abs) ^(B), it is not certain that theadiabatic-κ (EIT-like) system will always perform better than theconstant-κ one, in terms of the transferred and radiated energies. Infact, there could exist some range of the parameters (κ, Γ_(rad) ^(A),Γ_(rad) ^(B), Γ_(abs) ^(A), Γ_(abs) ^(B)), for which the energy radiatedin the constant-κ transfer case is less than that radiated in theEIT-like case. For this reason, we investigate both the adiabatic-κ andconstant-κ transfer schemes, as we vary some of the crucial parametersof the system. The percentage of energies transferred and lost(radiated+absorbed) depends only on the relative values of κ,Γ_(A)=Γ_(rad) ^(A)+Γ_(abs) ^(A) and Γ_(B)=Γ_(rad) ^(B)+Γ_(abs) ^(B).Hence we first calculate and visualize the dependence of these energieson the relevant parameters κ/Γ_(B) and Γ_(B)/Γ_(A), in the contour plotsshown in FIG. 21.

The way the contour plots are calculated is as follows. For each valueof (κ/Γ_(B), Γ_(B)/Γ_(A)) in the adiabatic case, where κ_(1B)(t) andκ_(B2)(t) are given by Eq. (44)-(45), one tries range of values oft_(EIT). For each t_(EIT), the maximum energy transferred E₂ (%) over0<t<t_(EIT), denoted by max(E₂, t_(EIT)), is calculated together withthe total energy lost at that maximum transfer. Next the maximum ofmax(E₂, t_(EIT)) over all values of t_(EIT) is selected and plotted assingle point on the contour plot in FIG. 21 a. We refer to this point asthe optimum energy transfer (%) in the adiabatic-κ case for theparticular (κ/Γ_(B), Γ_(B)/Γ_(A)) under consideration. We also plot inFIG. 21 d the corresponding value of the total energy lost (%) at theoptimum of E₂. We repeat these calculations for all pairs (κ/Γ_(B),Γ_(B)/Γ_(A)) shown in the contour plots. In the constant-κ transfercase, for each (κ/Γ_(B), Γ_(B)/Γ_(A)), the time evolution of E₂(%) andE_(lost) are calculated for 0<t<2/κ, and optimum transfer, shown in FIG.21 b, refers to the maximum of E₃(t) over 0<t<2/κ. The correspondingtotal energy lost at optimum constant-transfer is shown in FIG. 21 e.Now that we calculated the energies of interest as functions of(κ/Γ_(B), Γ_(B)/Γ_(A)), we look for ranges of the relevant parameters inwhich the adiabatic-κ transfer has advantages over the constant-κ one.So, we plot the ratio of (E₂)_(adiabatic-κ)/(E₂)_(constant-κ) in FIG. 21c, and (E_(lost))_(constant-κ)/(E_(lost))_(adiabatic-κ) in FIG. 21 f. Wefind that, for Γ_(B)/Γ_(A)>50, the optimum energy transferred in theadiabatic-κ case exceeds that in the constant-κ case, and theimprovement factor can be larger than 2. From FIG. 21 f, one sees thatthe adiabatic-κ scheme can reduce the total energy lost by factor of 3compared to the constant-κ scheme, also in the range Γ_(B)/Γ_(A)>50. Asin the constant-κ case, also in the adiabatic-κ case the efficiencyincreases as the ratio of the maximum value, κ, of the coupling rates tothe loss rate of the intermediate object (and thus also the first andsecond objects for Γ_(B)/Γ_(A)>1) increases. In some examples, one maydesign a system so that κ is larger than each of Γ_(B) and Γ_(A). Insome examples, one may design a system so that κ is at least 2 timeslarger than each of Γ_(B) and Γ_(A). In some examples, one may design asystem so that κ is at least 4 times larger than each of Γ_(B) andΓ_(A).

Although one may be interested in reducing the total energy lost(radiated+absorbed) as much as possible in order to make the transfermore efficient, the undersirable nature of the radiated energy may makeit important to consider reducing the energy radiated. For this purpose,we calculate the energy radiated at optimum transfer in both theadiabatic-κ and constant-κ schemes, and compare them. The relevantparameters in this case are κ/Γ_(B), Γ_(B)/Γ_(A), Γ_(rad) ^(A)/Γ_(A),and Γ_(rad) ^(B)/Γ_(B). The problem is more complex because theparameter space is now 4-dimensional. So we focus on those particularcross sections that can best reveal the most important differencesbetween the two schemes. From FIGS. 21 c and 21 f, one can guess thatthe best improvement in both E₂ and E_(lost) occurs for Γ_(B)/Γ_(A)≧500.Moreover, knowing that it is the intermediate object R_(B) that makesthe main difference between the adiabatic-κ and constant-κ schemes,being “energy-empty” in the adiabatic-κ case and “energy-full” in theconstant-κ one, we first look at the special situation where Γ_(rad)^(A)=0. In FIG. 22 a and FIG. 22 b, we show contour plots of the energyradiated at optimum transfer, in the constant-κ and adiabatic-κ schemesrespectively, for the particular cross section having Γ_(B)/Γ_(A)=500and Γ_(rad) ^(A)=0. Comparing these two figures, one can see that, byusing the adiabatic-κ scheme, one can reduce the energy radiated byfactor of 6.3 or more.

To get quantitative estimate of the radiation reduction factor in thegeneral case where Γ_(A,rad)≠0, we calculate the ratio of energiesradiated at optimum transfers in both schemes, namely,

$\begin{matrix}{\frac{\left( E_{rad} \right)_{{constant} - \kappa}}{\left( E_{rad} \right)_{{adiabatic} - \kappa}} = \frac{2{\int_{0}^{t_{*}^{{constant} - \kappa}}\left\{ {{\frac{\Gamma_{rad}^{B}}{\Gamma_{rad}^{A}}{{a_{B}^{{constant} - \kappa}(t)}}^{2}} + {{a_{1}^{{constant} - \kappa}(t)}}^{2} + {{a_{2}^{{constant} - \kappa}(t)}}^{2}} \right\}}}{2{\int_{0}^{t_{*}^{{adiabatic} - \kappa}}\left\{ {{\frac{\Gamma_{rad}^{B}}{\Gamma_{rad}^{A}}{{a_{B}^{{adiabatic} - \kappa}(t)}}^{2}} + {{a_{1}^{{adiabatic} - \kappa}(t)}}^{2} + {{a_{2}^{{adiabatic} - \kappa}(t)}}^{2}} \right\}}}} & (46)\end{matrix}$which depends only on Γ_(rad) ^(B)/Γ_(rad) ^(A), the time-dependent modeamplitudes, and the optimum transfer times in both schemes. The lattertwo quantities are completely determined by κ/Γ_(B) and Γ_(B)/Γ_(A).Hence the only parameters relevant to the calculations of the ratio ofradiated energies are Γ_(rad) ^(B)/Γ_(rad) ^(A), κ/Γ_(B) andΓ_(B)/Γ_(A), thus reducing the dimensionality of the investigatedparameter space from down to 3. For convenience, we multiply the firstrelevant parameter Γ_(rad) ^(B)/Γ_(rad) ^(A) by Γ_(B)/Γ_(A) whichbecomes (Γ_(rad) ^(B)/Γ_(B))/(Γ_(rad) ^(A)/Γ_(A)), i.e. the ratio ofquantities that specify what percentage of each object's loss isradiated. Next, we calculate the ratio of energies radiated as functionof (Γ_(rad) ^(B)/Γ_(B))/(Γ_(rad) ^(A)/Γ_(A)) and κ/Γ_(B) in the twospecial cases Γ_(B)/Γ_(A)=50, and Γ_(B)/Γ_(A)=500, and plot them in FIG.22 c and FIG. 22 d, respectively. We also show, in FIG. 22 e, thedependence of (E_(rad))_(constant-κ)/(E_(rad))_(EIT-like) on κ/Γ_(B) andΓ_(B)/Γ_(A), for the special case Γ_(rad) ^(A)=0. As can be seen fromFIG. 22 c-22 d, the adiabatic-κ scheme is less radiative than theconstant-κ scheme whenever Γ_(rad) ^(B)/Γ_(B) is larger than Γ_(rad)^(A)/Γ_(A), and the radiation reduction ratio increases as Γ_(B)/Γ_(A)and κ/Γ_(B) are increased (see FIG. 22 e). In some examples, theadiabatic-κ scheme is less radiative than the constant-κ scheme wheneverΓ_(rad) ^(B)/Γ_(rad) ^(A) is larger than about 20. In some examples, theadiabatic-κ scheme is less radiative than the constant-κ scheme wheneverΓ_(rad) ^(B)/Γ_(rad) ^(A) is larger than about 50.

It is to be understood that while three resonant objects are shown inthe previous examples, other examples can feature four or more resonantobjects. For example, in some examples, a single source object cantransfer energy to multiple device objects through one intermediateobject. In some examples, energy can be transferred from a sourceresonant object to a device resonant object, through two or moreintermediate resonant objects, and so forth.

6. System Sensitivity to Extraneous Objects

In general, the overall performance of an example of the resonance-basedwireless energy-transfer scheme depends strongly on the robustness ofthe resonant objects' resonances. Therefore, it is desirable to analyzethe resonant objects' sensitivity to the near presence of randomnon-resonant extraneous objects. One appropriate analytical model isthat of “perturbation theory” (PT), which suggests that in the presenceof an extraneous perturbing object p the field amplitude a₁(t) insidethe resonant object 1 satisfies, to first order:

$\begin{matrix}{\frac{\mathbb{d}a_{1}}{\mathbb{d}t} = {{{- {{\mathbb{i}}\left( {\omega_{1} - {{\mathbb{i}}\;\Gamma_{1}}} \right)}}a_{1}} + {{{\mathbb{i}}\left( {{\delta\;\kappa_{11{(p)}}} + {{\mathbb{i}}\;\delta\;\Gamma_{1{(p)}}}} \right)}a_{1}}}} & (47)\end{matrix}$where again ω₁ is the frequency and Γ₁ the intrinsic (absorption,radiation etc.) loss rate, while δκ_(11(p)) is the frequency shiftinduced onto 1 due to the presence of p and δΓ_(1(p)) is the extrinsicdue to p (absorption inside p, scattering from p etc.) loss rate.δΓ_(1(p)) is defined as δΓ_(1(p))≡Γ_(1(p))−Γ₁, where Γ_(1(p)) is thetotal perturbed loss rate in the presence of p. The first-order PT modelis valid only for small perturbations. Nevertheless, the parametersδκ_(11(p)), δΓ_(1(p)) are well defined, even outside that regime, if a₁is taken to be the amplitude of the exact perturbed mode. Note also thatinterference effects between the radiation field of the initialresonant-object mode and the field scattered off the extraneous objectcan for strong scattering (e.g. off metallic objects) result in totalΓ_(1,rad(p)) that are smaller than the initial Γ_(1,rad) (namelyδΓ_(1,rad(p)) is negative).

It has been shown that a specific relation is desired between theresonant frequencies of the source and device-objects and the drivingfrequency. In some examples, all resonant objects must have the sameeigenfrequency and this must be equal to the driving frequency. In someimplementations, this frequency-shift can be “fixed” by applying to oneor more resonant objects and the driving generator a feedback mechanismthat corrects their frequencies. In some examples, the driving frequencyfrom the generator can be fixed and only the resonant frequencies of theobjects can be tuned with respect to this driving frequency.

The resonant frequency of an object can be tuned by, for example,adjusting the geometric properties of the object (e.g. the height of aself-resonant coil, the capacitor plate spacing of a capacitively-loadedloop or coil, the dimensions of the inductor of an inductively-loadedrod, the shape of a dielectric disc, etc.) or changing the position of anon-resonant object in the vicinity of the resonant object.

In some examples, referring to FIG. 23 a, each resonant object isprovided with an oscillator at fixed frequency and a monitor whichdetermines the eigenfrequency of the object. At least one of theoscillator and the monitor is coupled to a frequency adjuster which canadjust the frequency of the resonant object. The frequency adjusterdetermines the difference between the fixed driving frequency and theobject frequency and acts, as described above, to bring the objectfrequency into the required relation with respect to the fixedfrequency. This technique assures that all resonant objects operate atthe same fixed frequency, even in the presence of extraneous objects.

In some examples, referring to FIG. 23( b), during energy transfer froma source object to a device object, the device object provides energy orpower to a load, and an efficiency monitor measures the efficiency ofthe energy-transfer or power-transmission. A frequency adjuster coupledto the load and the efficiency monitor acts, as described above, toadjust the frequency of the object to maximize the efficiency.

In other examples, the frequency adjusting scheme can rely oninformation exchange between the resonant objects. For example, thefrequency of a source object can be monitored and transmitted to adevice object, which is in turn synched to this frequency usingfrequency adjusters, as described above. In other embodiments thefrequency of a single clock can be transmitted to multiple devices, andeach device then synched to that frequency using frequency adjusters, asdescribed above.

Unlike the frequency shift, the extrinsic perturbing loss due to thepresence of extraneous perturbing objects can be detrimental to thefunctionality of the energy-transfer scheme, because it is difficult toremedy. Therefore, the total perturbed quality factors Q_((p)) (and thecorresponding perturbed strong-coupling factor U_((p)) should bequantified.

In some examples, a system for wireless energy-transfer uses primarilymagnetic resonances, wherein the energy stored in the near field in theair region surrounding the resonator is predominantly magnetic, whilethe electric energy is stored primarily inside the resonator. Suchresonances can exist in the quasi-static regime of operation (r<<λ) thatwe are considering: for example, for coils with h

2r, most of the electric field is localized within the self-capacitanceof the coil or the externally loading capacitor and, for dielectricdisks, with ∈>>1 the electric field is preferentially localized insidethe disk. In some examples, the influence of extraneous objects onmagnetic resonances is nearly absent. The reason is that extraneousnon-conducting objects p that could interact with the magnetic field inthe air region surrounding the resonator and act as a perturbation tothe resonance are those having significant magnetic properties (magneticpermeability Re{μ}>1 or magnetic loss Im{μ}>0). Since almost allevery-day non-conducting materials are non-magnetic but just dielectric,they respond to magnetic fields in the same way as free space, and thuswill not disturb the resonance of the resonator. Extraneous conductingmaterials can however lead to some extrinsic losses due to the eddycurrents induced inside them or on their surface (depending on theirconductivity). However, even for such conducting materials, theirpresence will not be detrimental to the resonances, as long as they arenot in very close proximity to the resonant objects.

The interaction between extraneous objects and resonant objects isreciprocal, namely, if an extraneous object does not influence aresonant object, then also the resonant object does not influence theextraneous object. This fact can be viewed in light of safetyconsiderations for human beings. Humans are also non-magnetic and cansustain strong magnetic fields without undergoing any risk. A typicalexample, where magnetic fields B˜1 T are safely used on humans, is theMagnetic Resonance Imaging (MRI) technique for medical testing. Incontrast, the magnetic near-field required in typical embodiments inorder to provide a few Watts of power to devices is only B˜10⁻⁴ T, whichis actually comparable to the magnitude of the Earth's magnetic field.Since, as explained above, a strong electric near-field is also notpresent and the radiation produced from this non-radiative scheme isminimal, the energy-transfer apparatus, methods and systems describedherein is believed safe for living organisms.

An advantage of the presently proposed technique using adiabaticvariations of the coupling rates between the first and intermediateresonators and between the intermediate and second resonators comparedto a mode of operation where these coupling rates are not varied but areconstant is that the interactions of the intermediate resonator withextraneous objects can be greatly reduced with the presently proposedscheme. The reason is once more the fact that there is always asubstantially small amount of energy in the intermediate resonator inthe adiabatic-κ scheme, therefore there is little energy that can beinduced from the intermediate object to an extraneous object in itsvicinity. Furthermore, since the losses of the intermediate resonatorare substantially avoided in the adiabatic-κ case, the system is lessimmune to potential reductions of the quality factor of the intermediateresonator due to extraneous objects in its vicinity.

6.1 Capacitively-Loaded Conducting Loops or Coils

In some examples, one can estimate the degree to which the resonantsystem of a capacitively-loaded conducting-wire coil has mostly magneticenergy stored in the space surrounding it. If one ignores the fringingelectric field from the capacitor, the electric and magnetic energydensities in the space surrounding the coil come just from the electricand magnetic field produced by the current in the wire; note that in thefar field, these two energy densities must be equal, as is always thecase for radiative fields. By using the results for the fields producedby a subwavelength (r

2) current loop (magnetic dipole) with h=0, we can calculate the ratioof electric to magnetic energy densities, as a function of distanceD_(p) from the center of the loop (in the limit r<<D_(p)) and the angleθ with respect to the loop axis:

$\begin{matrix}{{{\frac{w_{e}(x)}{w_{m}(x)} = {\frac{ɛ_{0}{{E(x)}}^{2}}{\mu_{0}{{H(x)}}^{2}} = \frac{\left( {1 + \frac{1}{x^{2}}} \right)\sin^{2}\theta}{{\left( {\frac{1}{x^{2}} + \frac{1}{x^{4}}} \right)4\;\cos^{2}\theta} + {\left( {1 - \frac{1}{x^{2}} + \frac{1}{x^{4}}} \right)\sin^{2}\theta}}}};}\mspace{20mu}{{x = {\left. {2\;\pi\frac{D_{p}}{\lambda}}\Rightarrow\frac{∯\limits_{S_{p}}{{w_{e}(x)}{\mathbb{d}S}}}{∯\limits_{S_{p}}{{w_{m}(x)}{\mathbb{d}S}}} \right. = \frac{1 + \frac{1}{x^{2}}}{1 + \frac{1}{x^{2}} + \frac{3}{x^{4}}}}};}\mspace{20mu}{{x = {2\;\pi\frac{D_{p}}{\lambda}}},}} & (48)\end{matrix}$where the second line is the ratio of averages over all angles byintegrating the electric and magnetic energy densities over the surfaceof a sphere of radius D_(p). From Eq. (48) it is obvious that indeed forall angles in the near field (x

1) the magnetic energy density is dominant, while in the far field (x

1) they are equal as they should be. Also, the preferred positioning ofthe loop is such that objects which can interfere with its resonance lieclose to its axis (θ=0), where there is no electric field. For example,using the systems described in Table 4, we can estimate from Eq. (48)that for the loop of r=30 cm at a distance D_(p)=10r=3 m the ratio ofaverage electric to average magnetic energy density would be ˜12% and atD_(p)=3r=90 cm it would be ˜1%, and for the loop of r=10 cm at adistance D_(p)=10r=1 m the ratio would be ˜33% and at D_(P)=3r=30 cm itwould be ˜2.5%. At closer distances this ratio is even smaller and thusthe energy is predominantly magnetic in the near field, while in theradiative far field, where they are necessarily of the same order(ratio→1), both are very small, because the fields have significantlydecayed, as capacitively-loaded coil systems are designed to radiatevery little. Therefore, this is the criterion that qualifies this classof resonant system as a magnetic resonant system.

To provide an estimate of the effect of extraneous objects on theresonance of a capacitively-loaded loop including the capacitor fringingelectric field, we use the perturbation theory formula, stated earlier,δΓ_(1,abs(p))=ω₁/4·∫d³r Im{∈_(p)(r)}|E₁(r)|²/W with the computationalFEFD results for the field of an example like the one shown in the plotof FIG. 5 and with a rectangular object of dimensions 30 cm×30 cm×1.5 mand permittivity s=49+16i (consistent with human muscles) residingbetween the loops and almost standing on top of one capacitor (˜3 cmaway from it) and find δQ_(abs(human))˜10⁵ and for ˜10 cm awayδQ_(abs(human))˜5·10⁵. Thus, for ordinary distances (˜1 m) andplacements (not immediately on top of the capacitor) or for mostordinary extraneous objects p of much smaller loss-tangent, we concludethat it is indeed fair to say that δQ_(abs(p))→∞. The only perturbationthat is expected to affect these resonances is a close proximity oflarge metallic structures.

Self-resonant coils can be more sensitive than capacitively-loadedcoils, since for the former the electric field extends over a muchlarger region in space (the entire coil) rather than for the latter(just inside the capacitor). On the other hand, self-resonant coils canbe simple to make and can withstand much larger voltages than mostlumped capacitors. Inductively-loaded conducting rods can also be moresensitive than capacitively-loaded coils, since they rely on theelectric field to achieve the coupling.

6.2 Dielectric Disks

For dielectric disks, small, low-index, low-material-loss or far-awaystray objects will induce small scattering and absorption. In such casesof small perturbations these extrinsic loss mechanisms can be quantifiedusing respectively the analytical first-order perturbation theoryformulas[δQ _(1,rad(p))]⁻¹≡2δΓ_(1,rad(p))/ω₁ ∝∫d ³ r[Re{∈ _(p)(r)}|E ₁(r)|]² /W[δQ _(1,abs(p))]⁻¹≡2δΓ_(1,abs(p))/ω₁ =∫d ³ rIm{∈ _(p)(r)}|E ₁(r)|²/2Wwhere W=∫d³r∈(r)|E₁(r)|²/2 is the total resonant electromagnetic energyof the unperturbed mode. As one can see, both of these losses depend onthe square of the resonant electric field tails E1 at the site of theextraneous object. In contrast, the coupling factor from object 1 toanother resonant object 2 is, as stated earlier,k ₁₂=2κ₁₂/√{square root over (ω₁ω₂)}≈∫d³ r∈ ₂(r)E ₂*(r)E ₁(r)/∫d ³r∈(r)|E ₁(r)|²and depends linearly on the field tails E₁ of 1 inside 2. Thisdifference in scaling gives us confidence that, for, for example,exponentially small field tails, coupling to other resonant objectsshould be much faster than all extrinsic loss rates (κ₁₂

δΓ_(1,2(p))), at least for small perturbations, and thus theenergy-transfer scheme is expected to be sturdy for this class ofresonant dielectric disks.

However, we also want to examine certain possible situations whereextraneous objects cause perturbations too strong to analyze using theabove first-order perturbation theory approach. For example, we place adielectric disk close to another off-resonance object of large Re{∈},Im{∈} and of same size but different shape (such as a human being h), asshown in FIG. 24 a, and a roughened surface of large extent but of smallRe{∈}, Im{∈} (such as a wall w), as shown in FIG. 24 b. For distancesD_(h,w)/r=10−3 between the disk-center and the “human”-center or “wall”,the numerical FDFD simulation results presented in FIGS. 24 a and 24 bsuggest that, the disk resonance seems to be fairly robust, since it isnot detrimentally disturbed by the presence of extraneous objects, withthe exception of the very close proximity of high-loss objects. Toexamine the influence of large perturbations on an entireenergy-transfer system we consider two resonant disks in the closepresence of both a “human” and a “wall”. Comparing Table 8 to the tablein FIG. 24 c, the numerical FDFD simulations show that the systemperformance deteriorates from U˜1-50 to U_((hw))˜0.5-10, i.e. only byacceptably small amounts.

In general, different examples of resonant systems have different degreeof sensitivity to external perturbations, and the resonant system ofchoice depends on the particular application at hand, and how importantmatters of sensitivity or safety are for that application. For example,for a medical implantable device (such as a wirelessly poweredartificial heart) the electric field extent must be minimized to thehighest degree possible to protect the tissue surrounding the device. Insuch cases where sensitivity to external objects or safety is important,one should design the resonant systems so that the ratio of electric tomagnetic energy density w_(e)/w_(m) is reduced or minimized at most ofthe desired (according to the application) points in the surroundingspace.

7 Applications

The non-radiative wireless energy transfer techniques described abovecan enable efficient wireless energy-exchange between resonant objects,while suffering only modest transfer and dissipation of energy intoother extraneous off-resonant objects. The technique is general, and canbe applied to a variety of resonant systems in nature. In this Section,we identify a variety of applications that can benefit from or bedesigned to utilize wireless power transmission.

Remote devices can be powered directly, using the wirelessly suppliedpower or energy to operate or run the devices, or the devices can bepowered by or through or in addition to a battery or energy storageunit, where the battery is occasionally being charged or re-chargedwirelessly. The devices can be powered by hybrid battery/energy storagedevices such as batteries with integrated storage capacitors and thelike. Furthermore, novel battery and energy storage devices can bedesigned to take advantage of the operational improvements enabled bywireless power transmission systems.

Devices can be turned off and the wirelessly supplied power or energyused to charge or recharge a battery or energy storage unit. The batteryor energy storage unit charging or recharging rate can be high or low.The battery or energy storage units can be trickle charged or floatcharged. It would be understood by one of ordinary skill in the art thatthere are a variety of ways to power and/or charge devices, and thevariety of ways could be applied to the list of applications thatfollows.

Some wireless energy transfer examples that can have a variety ofpossible applications include for example, placing a source (e.g. oneconnected to the wired electricity network) on the ceiling of a room,while devices such as robots, vehicles, computers, PDAs or similar areplaced or move freely within the room. Other applications can includepowering or recharging electric-engine buses and/or hybrid cars andmedical implantable devices. Additional example applications include theability to power or recharge autonomous electronics (e.g. laptops,cell-phones, portable music players, house-hold robots, GPS navigationsystems, displays, etc), sensors, industrial and manufacturingequipment, medical devices and monitors, home appliances (e.g. lights,fans, heaters, displays, televisions, counter-top appliances, etc.),military devices, heated or illuminated clothing, communications andnavigation equipment, including equipment built into vehicles, clothingand protective-wear such as helmets, body armor and vests, and the like,and the ability to transmit power to physically isolated devices such asto implanted medical devices, to hidden, buried, implanted or embeddedsensors or tags, to and/or from roof-top solar panels to indoordistribution panels, and the like.

A number of examples of the invention have been described. Nevertheless,it will be understood that various modifications can be made withoutdeparting from the spirit and scope of the invention.

What is claimed is:
 1. A method for transferring energy wirelessly, the method comprising: transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_(1B); transferring energy wirelessly from the intermediate resonator structure to a second resonator structure, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ_(B2); and during the wireless energy transfers, adjusting at least one of the coupling rates κ_(1B) and κ_(B2) as a function of time to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.
 2. The method of claim 1, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) minimizes energy accumulation in the intermediate resonator structure and causes wireless energy transfer from the first resonator structure to the second resonator structure.
 3. The method of claim 1, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) maintains energy distribution in the field of the three-resonator system in an eigenstate having substantially no energy in the intermediate resonator structure.
 4. The method of claim 3, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) further causes the eigenstate to evolve substantially adiabatically from an initial state with substantially all energy in the resonator structures in the first resonator structure to a final state with substantially all of the energy in the resonator structures in the second resonator structure.
 5. The method of claim 1, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) comprises adjustments of both coupling rates κ_(1B) and κ_(B2) during wireless energy transfer.
 6. The method of claim 1, wherein the resonator structures each have a quality factor larger than
 10. 7. The method of claim 1, wherein resonant energy in each of the resonator structures comprises electromagnetic fields.
 8. The method of claim 7, wherein the maximum value of the coupling rate κ_(1B) and the maximum value of the coupling rate κ_(B2) for inductive coupling between the intermediate resonator structure and each of the first and second resonator structures are each larger than twice the loss rate Γ for each of the first and second resonators.
 9. The method of claim 8, wherein the maximum value of the coupling rate κ_(1B) and the maximum value of the coupling rate κ_(B2) for inductive coupling between the intermediate resonator structure and each of the first and second resonator structures are each larger than four (4) times the loss rate Γ for each of the first and second resonators.
 10. The method of claim 7, wherein each resonator structure has a resonant frequency between 50 KHz and 500 MHz.
 11. The method of claim 1, wherein the maximum value of the coupling rate κ_(1B) and the maximum value of the coupling rate κ_(B2) are each at least five (5) times greater than the coupling rate between the first resonator structure and the second resonator structure.
 12. The method of claim 1, wherein the intermediate resonator structure has a rate of radiative energy loss that is at least twenty (20) times greater than that for either the first resonator structure or the second resonator structure.
 13. The method of claim 1, wherein the first and second resonator structures are substantially identical.
 14. The method of claim 1, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) causes peak energy accumulation in the intermediate resonator structure to be less than five percent (5%) of the peak total energy in the three resonator structures.
 15. The method of claim 1, wherein adjusting at least one of the coupling rates κ_(1B) and κ_(B2) comprises adjusting a relative position and/or orientation between one or more pairs of the resonator structures.
 16. The method of claim 1, wherein adjusting at least one of the coupling rates κ_(1B) and κ_(B2) comprises adjusting a resonator property of one or more of the resonator structures.
 17. The method of claim 16, wherein the resonator property comprises mutual inductance.
 18. The method of claim 1, wherein at least one of the resonator structures comprises a capacitively loaded loop or coil of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
 19. The method of claim 1, wherein at least one of the resonator structures comprises an inductively loaded rod of at least one of a conducting wire, a conducting Litz wire, and a conducting ribbon.
 20. The method of claim 4, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) causes peak energy accumulation in the intermediate resonator structure during the wireless energy transfers to be less than ten percent (10%) of the peak total energy in the three resonator structures.
 21. The method of claim 1, wherein the wireless energy transfers are non-radiative energy transfers mediated by a coupling of a resonant field evanescent tail of the first resonator structure and a resonant field evanescent tail of the intermediate resonator structure and a coupling of the resonant field evanescent tail of the intermediate resonator structure and a resonant field evanescent tail of the second resonator structure.
 22. The method of claim 1, wherein the first and second resonator structures each have a quality factor greater than
 50. 23. The method of claim 1, wherein the first and second resonator structures each have a quality factor greater than
 100. 24. The method of claim 1, wherein during the wireless energy transfer from the first resonator structure to the second resonator structure at least one of the coupling rates is adjusted so that κ_(1B)<<K_(B2) at a start of the energy transfer and κ_(1B)>>κ_(B2) by a time a substantial portion of the energy has been transferred from the first resonator structure to the second resonator structure.
 25. A method for transferring energy wirelessly, the method comprising: transferring energy wirelessly from a first resonator structure to an intermediate resonator structure, wherein the coupling rate between the first resonator structure and the intermediate resonator structure is κ_(1B); transferring energy wirelessly from the intermediate resonator structure to a second resonator, wherein the coupling rate between the intermediate resonator structure and the second resonator structure is κ_(B2); and during the wireless energy transfers, adjusting at least one of the coupling rates κ_(1B) and κ_(B2) to cause an energy distribution in the field of the three-resonator system to have substantially no energy in the intermediate resonator structure while wirelessly transferring energy from the first resonator structure to the second resonator structure through the intermediate resonator structure.
 26. The method of claim 25, wherein having substantially no energy in the intermediate resonator structure means that peak energy accumulation in the intermediate resonator structure is less than ten percent (10%) of the peak total energy in the three resonator structures throughout the wireless energy transfer.
 27. The method of claim 25, wherein having substantially no energy in the intermediate resonator structure means that peak energy accumulation in the intermediate resonator structure is less than five percent (5%) of the peak total energy in the three resonator structures throughout the wireless energy transfer.
 28. The method of claim 25, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) maintains the energy distribution in the field of the three-resonator system in an eigenstate having the substantially no energy in the intermediate resonator structure.
 29. The method of claim 28, wherein the adjustment of at least one of the coupling rates κ_(1B) and κ_(B2) further causes the eigenstate to evolve substantially adiabatically from an initial state with substantially all energy in the resonator structures in the first resonator structure to a final state with substantially all of the energy in the resonator structures in the second resonator structure.
 30. An apparatus comprising: first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ_(1B) and a coupling rate between the intermediate resonator structure and the second resonator structure is κ_(B2); and means for adjusting at least one of the coupling rates κ_(1B) and κ_(B2) during wireless energy transfers among the resonator structures to cause an energy distribution in the field of the three-resonator system to have substantially no energy in the intermediate resonator structure while wirelessly transferring energy from the first resonator structure to the second resonator structure through the intermediate resonator structure.
 31. An apparatus comprising: first, intermediate, and second resonator structures, wherein a coupling rate between the first resonator structure and the intermediate resonator structure is κ_(1B) and a coupling rate between the intermediate resonator structure and the second resonator structure is κ_(B2); and means for adjusting at least one of the coupling rates κ_(1B) and κ_(B2) during wireless energy transfers among the resonator structures as a function of time to reduce energy accumulation in the intermediate resonator structure and improve wireless energy transfer from the first resonator structure to the second resonator structure through the intermediate resonator structure.
 32. The apparatus of claim 31, wherein the means for adjusting comprises a rotation stage for adjusting the relative orientation of the intermediate resonator structure with respect to the first and second resonator structures.
 33. The apparatus of claim 31, wherein the means for adjusting comprises a translation stage for moving the first and/or second resonator structures relative to the intermediate resonator structure.
 34. The apparatus of claim 31, wherein the means for adjusting comprises a mechanical, electro-mechanical, or electrical staging system for dynamically adjusting the effective size of one or more of the resonator structures. 